Dynamic Network User Equilibrium (Complex Networks and Dynamic Systems, 5) 3031255623, 9783031255625

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Complex Networks and Dynamic Systems 5

Terry L. Friesz Ke Han

Dynamic Network User Equilibrium

Complex Networks and Dynamic Systems Volume 5

Series Editor Terry L. Friesz, University Park, Pennsylvania State University, State College, PA, USA

Networks have become the focus of intense interdisciplinary research. There is a growing consensus that social, economic and engineered networks are naturally coupled and that their joint study, using the tools of complexity science, holds great transformative potential. Around the globe, scholars of many backgrounds are finding they have common interests in understanding, manipulating, and exploiting the properties of networks. The objects of study by this scientific movement are network depictions of multiagent systems characterized by complex interactions occurring over space and time. The networks studied include transportation, electric power, water distribution and telecommunications grids; disease transmission, terrorism and social networks; ecological networks; economic and financial transaction networks; and the Internet. In response to such network-oriented scholarly inquiry, a new book series entitled Complex Networks and Dynamic Systems has been launched to help propel as well as document the huge pulse of social, economic and engineering network research that is now underway. As such the series is intended to be of interest to economists, geographers, regional scientists, civil engineers, city planners, industrial engineers, operations researchers, and financial engineers. High quality original research monographs, as well as conference proceedings and primers, are now being actively considered for publication. Potential authors and proceedings editors are encouraged to contact the Series Editor, Professor T. L. Friesz ([email protected]).

Terry L. Friesz • Ke Han

Dynamic Network User Equilibrium

Terry L. Friesz Department of Industrial & Manufacturing Engineering The Pennsylvania State University University Park, PA, USA

Ke Han School of Transportation and Logistics Southwest Jiaotong University Chengdu, China

ISSN 2195-724X ISSN 2195-7258 (electronic) Complex Networks and Dynamic Systems ISBN 978-3-031-25562-5 ISBN 978-3-031-25564-9 (eBook) https://doi.org/10.1007/978-3-031-25564-9 © Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book is about a relatively new field of scholarly inquiry usually referred to as dynamic network user equilibrium, now almost universally abbreviated as DUE. The problem known as DUE is a special instance of dynamic traffic assignment (DTA), which is the descriptive modeling of time-varying flows on urban vehicular networks. DUE views urban traffic flows as the result of a Nash-like noncooperative differential game that has a variety of mathematical forms and has been the subject of study of hundreds of research manuscripts. It has only recently been understood that many of the DUE models reported to date can be reformulated as a so-called differential variational inequality (DVI). Articulation of DUE as a DVI offers advantages for both qualitative and numerical analyses of differential noncooperative games that describe urban network flows. We provide in this book the first synthesis of results obtained over the last decade from use of the DVI formalism to study the DUE problem. In particular, we explore the intimately related problem of dynamic network loading that determines the arc flows and effective travel delays (or generalized travel costs) arising from the expression of departure rates at the origins of commuter trips between home and workplace. In particular, we show that dynamic network loading with spillback of queues into upstream arcs may be formulated as a differential algebraic system. We also show how the dynamic network loading problem and the dynamic user equilibrium problem may be solved simultaneously rather than sequentially. We present two distinct categories of network loading problems-those based on Vickrey’s congestion bottleneck problem and those based on the hydrodynamic theory of traffic flow. We show how the first-in-first-out queue discipline may be maintained for each. A number of recent and new extensions of the DVI-based theory of DUE are presented. In particular, the fixed travel demand assumption that has preoccupied DUE modeling is replaced by an elastic demand formulation of DUE. In turn, the elastic demand formulation is extended to consider bounded rationality, wherein travelers are willing to undertake travel that involves less than optimal outcomes provided the extent of suboptimality is bounded.

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The network loading and user equilibrium aspects of DUE modeling are employed to articulate a fixed-point algorithm that computes dynamic user equilibria. We note that the fixed point algorithm possesses subproblems that are linearquadratic optimal control problems, which may be solved with great efficiency. We discuss conditions that assure the convergence of the fixed point algorithm. Other algorithms are also discussed. Several numerical examples are presented and discussed. Relevant mathematical background material is provided to make the book as widely accessible as possible. University Park, PA, USA London, UK

Terry L. Friesz Ke Han

Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Some DTA and DUE Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Vocabulary of DUE Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Alternative Formulations of DUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Structure of DUE Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Dynamic Network Loading Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Vickrey’s Model of a Traffic Bottleneck and Its Extension . . . 1.6.2 Network Loading Based on Link Dynamics . . . . . . . . . . . . . . . . . . 1.6.3 Network Loading as a LWR-Based PDAE System . . . . . . . . . . . 1.6.3.1 The Perakis-Kachani DNL Models . . . . . . . . . . . . . . . . . . 1.6.3.2 The Han-Friesz Within-Link Dynamics for DNL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4 Network Loading Based on the CTM. . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.5 Network Loading Based on the Variational Approach . . . . . . . . 1.6.6 LWR Network Loading Based on Closed-Form Operators . . . 1.6.7 Dynamic User Equilibrium in Continuous Time . . . . . . . . . . . . . . 1.6.8 Dynamic User Equilibrium in Discrete Time . . . . . . . . . . . . . . . . . 1.7 Other Considerations in Classifying DUE Models . . . . . . . . . . . . . . . . . . . . 1.8 Unresolved/Partially Resolved Fundamental Challenges . . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 6 7 9 10 12 17 17

Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Selected Topics in Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Hilbert Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 The Contraction Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Nonlinear Program Defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Fritz John Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 33 35 36 38 39 39 40

19 21 22 23 23 25 26 26 27

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2.2.3 The Kuhn-Tucker Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Kuhn-Tucker Conditions Sufficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Kuhn-Tucker Conditions for Variational Inequalities . . . . . . . . . 2.3 Calculus of Variations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The Space C 1 [t0 , tf ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The Concept of a Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 The State Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Necessary Conditions for Continuous-Time Optimal Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Sufficiency in Optimal Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.1 The Mangasarian Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.2 The Arrow Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Differential Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Regularity Conditions for DIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Necessary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Nash Games and Differential Nash Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Nash Equilibria and Normal Form Games . . . . . . . . . . . . . . . . . . . . 2.6.2 Differential Nash Games and Differential Nash Equilibria . . . 2.6.3 Generalized Differential Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . 2.7 The Scalar Conservation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Definition and Examples of the Scalar Conservation Law. . . . 2.7.2 Characteristics, Shock Waves, and Weak Solutions . . . . . . . . . . . 2.7.3 Non-uniqueness of Integral Solutions, Entropy Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 The Hamilton-Jacobi Equations and the Variational Principle. . . . . . . . 2.8.1 The Hamilton-Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 The Variational Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2.1 The Classical Lax-Hopf Formula . . . . . . . . . . . . . . . . . . . . 2.8.2.2 The Generalized Lax-Hopf Formula . . . . . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

40 41 43 45 45 46 48 50 53 57 57 59 61 62 63 64 66 66 68 72 73 74 76 80 83 83 84 84 85 89

The Variational Inequality Formulation of Dynamic User Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.1 Notation and Essential Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.2 The VI Formulation of DUE with Fixed Demand . . . . . . . . . . . . . . . . . . . . . 95 3.2.1 Definition of DUE with Fixed Demand. . . . . . . . . . . . . . . . . . . . . . . . 95 3.2.2 Variational Inequality Problem Equivalent to DUE with Fixed Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.3 The VI Formulation of DUE with Elastic Demand. . . . . . . . . . . . . . . . . . . . 101 3.3.1 Definition of DUE with Elastic Demand . . . . . . . . . . . . . . . . . . . . . . 101 3.3.2 Variational Inequality Problem Equivalent to DUE with Elastic Demand. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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3.4 The VI Formulation of Boundedly Rational DUE. . . . . . . . . . . . . . . . . . . . . 3.4.1 Definition of DUE with Bounded Rationality . . . . . . . . . . . . . . . . . 3.4.2 Variational Inequality Equivalent to BR-DUE with Exogenous or Endogenous Tolerances . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Kuhn-Tucker Conditions for DUE Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Application to DUE with Fixed Demand . . . . . . . . . . . . . . . . . . . . . . 3.5.1.1 Kuhn-Tucker Conditions for Discrete-time DUE Problems with Fixed Demand . . . . . . . . . . . . . . . . . 3.5.1.2 An Equivalent Linear System . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Application to DUE with Elastic Demand. . . . . . . . . . . . . . . . . . . . . 3.5.2.1 Kuhn-Tucker Conditions for Discrete-time DUE Problems with Elastic Demand . . . . . . . . . . . . . . . . 3.5.2.2 An Equivalent Linear System . . . . . . . . . . . . . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5

The Differential Variational Inequality Formulation of Dynamic User Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 DVI Formulation of DUE with Fixed Demand . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fixed-Point Problem Formulation of DUE with Fixed Demand. . . . . . 4.3 DVI Formulation of DUE with Elastic Demand . . . . . . . . . . . . . . . . . . . . . . . 4.4 Fixed-Point Problem Formulation of DUE with Elastic Demand . . . . 4.5 DVI Formulation of DUE with Bounded Rationality . . . . . . . . . . . . . . . . . 4.5.1 With Exogenous Tolerances (BR-DUE) . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 With Endogenous Tolerances (VT-BR-DUE) . . . . . . . . . . . . . . . . . 4.6 Fixed-Point Problem Formulation of DUE with Bounded Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 With Exogenous Tolerances (BR-DUE) . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 With Endogenous Tolerances (VT-BR-DUE) . . . . . . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Existence of Dynamic User Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Existence of Dynamic User Equilibrium with Fixed Demand . . . . . . . . 5.1.1 Mathematical Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Alternative Expression of the Effective Path Delay . . . . . . . . . . . 5.1.3 The Existence Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Existence of Dynamic User Equilibrium with Elastic Demand. . . . . . . 5.2.1 The Variational Inequality Formulation in an Extended Hilbert Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 The Existence Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 An Example of Non-existence of DUE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Existence of Dynamic User Equilibrium with Bounded Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Analytical Properties of the New Operator . . . . . . . . . . . . . . . . . . . . 5.4.2 The Existence Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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106 108 110 115 115 115 118 118 119 121 122 125 126 129 131 136 139 140 144 148 148 150 153 155 157 157 159 161 167 167 169 173 177 178 183

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5.5 Characterization of Solutions for Dynamic User Equilibrium with Bounded Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Discrete-Time VT-BR-DUE Problem . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Characterization of the Solution Set of VT-BR-DUE . . . . . . . . . 5.5.3 Constructing Connected Subset of the Solution Set . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

7

185 186 189 195 197

Algorithms for Computing Dynamic User Equilibria . . . . . . . . . . . . . . . . . . . 6.1 Fixed-Point Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Fixed-Point Algorithm for DUE with Fixed Demand. . . . . . . . . 6.1.2 Fixed-Point Algorithm for DUE with Elastic Demand . . . . . . . 6.1.3 Fixed-Point Algorithm for DUE with Bounded Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Self-adaptive Projection Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Self-adaptive Projection Algorithm for DUE with Fixed Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Self-adaptive Projection Algorithm for DUE with Elastic Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Self-adaptive Projection Algorithm for DUE with Bounded Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Proximal Point Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Proximal Point Method for DUE with Fixed Demand . . . . . . . . 6.3.2 Proximal Point Method for DUE with Elastic Demand. . . . . . . 6.3.3 Proximal Point Method for DUE with Bounded Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Convergence of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Convergence Result for the Fixed-Point Algorithm . . . . . . . . . . . 6.4.1.1 A Generic Proof Based on Strong Monotonicity . . . 6.4.1.2 An Adapted Proof for E-DUE Based on Weak Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Convergence Result for the Self-adaptive Projection Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Convergence Result for the Proximal Point Method . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 199 199 202

Dynamic Network Loading: Non-physical Queue Models . . . . . . . . . . . . . . 7.1 The Link Delay Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Link Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Network Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Formulation of the Dynamic Network Loading Problem . . . . . 7.1.4 Continuity of the Effective Path Delay Operator . . . . . . . . . . . . . . 7.2 The Vickrey Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Ordinary Differential Equation Formulation . . . . . . . . . . . . . . . . . . 7.2.1.1 The Linear Complementarity System Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1.2 The α-Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

243 244 244 248 250 252 259 260

207 211 211 213 216 218 218 220 223 225 226 226 227 233 238 241

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7.2.1.3 The ε-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Differential Equation Formulation . . . . . . . . . . . . . . . . . . . . . Closed-Form Solutions of the Vickrey Model . . . . . . . . . . . . . . . . . The Generalized Vickrey Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4.1 Initial-Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . 7.2.5 DAE System Formulation of Dynamic Network Loading . . . . 7.2.6 Continuity of the Effective Path Delay Operator . . . . . . . . . . . . . . 7.3 Some Other Non-physical Queue Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Models Based on Link Exit Flow Functions . . . . . . . . . . . . . . . . . . 7.3.2 Models with Controlled Entrance and Exit Flows . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263 264 266 276 277 281 284 289 289 289 291

Dynamic Network Loading: Physical Queue Models . . . . . . . . . . . . . . . . . . . . 8.1 The Lighthill-Whitham-Richards Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 The LWR Model at Road Junctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1.1 Demand and Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1.2 Riemann Solver for the Diverge Junction . . . . . . . . . . . 8.1.1.3 Riemann Solver for the Merge Junction. . . . . . . . . . . . . 8.1.2 The LWR Model on Road Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Partial Differential Algebraic Equation System Formulation of Dynamic Network Loading . . . . . . . . . . . . . . . . . . . 8.1.3.1 Within-Link Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3.2 Determination of Boundary Conditions at an Ordinary Node . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3.3 Determination of Flow Distribution at Origin/Destination Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3.4 Calculation of Path Travel Times . . . . . . . . . . . . . . . . . . . . 8.1.3.5 The PDAE System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Variational Formulation of the LWR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Notation and Brief Recap of Variational Theory . . . . . . . . . . . . . . 8.2.2 Dynamics at the Origin Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Differential Algebraic Equation System Formulation with General Fundamental Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3.1 Determination of Link Demand and Supply . . . . . . . . 8.2.3.2 Determination of Boundary Conditions Using Demand and Supply. . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3.3 The DAE System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Differential Algebraic Equation System Formulation with Triangular Fundamental Diagram . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4.1 Determination of Link Demand and Supply . . . . . . . . 8.2.4.2 The DAE System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Cell Transmission Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Link Dynamic of the CTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Network Extension of the CTM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2.1 Ordinary Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

295 295 296 299 300 301 303

7.2.2 7.2.3 7.2.4

8

304 305 305 307 309 309 311 311 312 313 314 316 317 318 318 319 320 320 322 322

xii

9

Contents

8.3.2.2 Merge Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2.3 Diverge Junction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 The Link Transmission Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Link Dynamics of the LTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Network Extension of the LTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Relationship with the Continuous-Time Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Continuity of the Effective Delay Operator for LWR-Based Dynamic Network Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Well-Posedness of Junction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1.1 An Example of Ill-posedness . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1.2 Generalized Tangent Vectors. . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1.3 A Sufficient Condition for the Well-Posedness of the Diverge Model. . . . . . . . . . . . . . . 8.5.2 An Estimation of Minimum Network Supply . . . . . . . . . . . . . . . . . 8.5.3 The Continuity Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3.1 Estimates Regarding the Path Disaggregation Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3.2 Well-Posedness of the Queuing Model at the Origin with Respect to Departure Rates. . . . . . . . . 8.5.3.3 The Continuity of the Path Delay Operator . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

323 324 325 325 326

Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Closed-Form Solutions of Dynamic User Equilibria . . . . . . . . . . . . . . . . . . 9.1.1 Simultaneous Route-and-Departure-Time DUE with Fixed Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Kuhn-Tucker Conditions for the Fixed Demand DUE . . . . . . . . 9.1.3 Simultaneous Route-and-Departure-Time DUE with Elastic Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Kuhn-Tucker Conditions for the Elastic Demand DUE. . . . . . . 9.2 Basic Setup of Numerical Examples of Dynamic User Equilibria . . . 9.3 Numerical Solutions of DUE with Fixed Demand . . . . . . . . . . . . . . . . . . . . 9.3.1 Performance of the Fixed-Point Algorithm. . . . . . . . . . . . . . . . . . . . 9.3.2 DUE Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Numerical Examples of DUE with Elastic Demand. . . . . . . . . . . . . . . . . . . 9.4.1 The Seven-Link Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 The Sioux Falls Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Algorithm Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Numerical Examples of DUE with Bounded Rationality . . . . . . . . . . . . . 9.5.1 VT-BR-DUE on a Seven-Link Network . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 BR-DUE on the Nguyen Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 BR-DUE on the Sioux Falls Network . . . . . . . . . . . . . . . . . . . . . . . . . References and Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

353 353

327 329 329 330 332 335 339 344 344 345 347 348

355 360 362 368 370 371 371 373 376 376 379 380 381 381 384 385 388

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Chapter 1

Introduction

1.1 Introduction As a field of inquiry, dynamic traffic assignment (DTA) is almost always understood to mean mathematical modeling and computational methods for describing and predicting time-varying flows on vehicular traffic networks. The field is aptly further characterized by saying it is rapidly changing and steadily becoming more realitybased. In fact, DTA scholarship is just now showing the first signs of its potential applicability and computability for the planning and control of real vehicular networks. Dynamic traffic assignment has to date focused on appropriate extensions of the notions of system optimal flows and user equilibrium inherited from the theory of static traffic equilibrium, generally credited to Wardrop (1952). As taught in introductory courses on traffic science, system optimal (SO) flows devolve from Wardrop’s so-called second principle, which demands that total congestion for the network of interest be minimized. By contrast, Wardrop’s first principle is concerned with the so-called user equilibrium (UE) flow patterns resulting from noncooperative competition among agents for the limited capacity of the network of interest. As such, a user equilibrium is a flow pattern consistent with Wardrop’s first principle. Moreover, it has the essential characteristic that unit travel costs on paths joining the same origin-destination pair are equal. Extension of the static equilibrium notions of system optimal and user equilibrium traffic assignment to a dynamic setting has been the main focus of DTA research since the seminal papers by Merchant and Nemhauser (1978a,b). We can say that presently there is a large body of literature on dynamic system optimal (DSO) and dynamic user equilibrium (DUE) models that are widely referred to as analytical models of dynamic traffic assignment. Our presentation will illustrate these models are similar yet differ in very important details. Perhaps the greatest similarity found among DUE models is that, with few exceptions, they take an open-loop perspective wherein departure rates, departure times, route choice, travel delays, and arrival times are computed by assuming all agents have perfect, © Springer Nature Switzerland AG 2022 T. L. Friesz, K. Han, Dynamic Network User Equilibrium, Complex Networks and Dynamic Systems 5, https://doi.org/10.1007/978-3-031-25564-9_1

1

2

1 Introduction

complete information; such agents may be viewed as mere actors in a play since, metaphorically speaking, all dramatic action and speech are predetermined. There are several fundamental modeling and computational issues that must be solved before DTA approached from the perspective of DUE may be considered a fully mature field of inquiry. It is easiest to express those considerations from the perspective of so-called analytical dynamic traffic assignment, wherein systems of operators, equations, inequalities, and extremal principles are employed to depict the behavior of agents and classes of agents engaged in route choice, route updating, and departure time choice in a potentially multi-timescale decision environment. In analytical DTA, all aspects of the models considered are mathematically articulated as allowed operations on a finite-dimensional or an infinite-dimensional closed vector subspace. Thus, analytical DTA does not generally include simulation models wherein behaviors are expressed via programming languages in ways that prevent exploration via functional analysis and operator theory.

1.2 Some DTA and DUE Literature The field of dynamic traffic assignment (DTA), as we have commented above, includes notions of dynamic user equilibrium. To date, there have been many papers and a few books that contain summaries of DTA and address, among other things, dynamic user equilibrium (DUE) modeling. In particular, we next discuss what we believe to be the most significant DTA papers that also review the intellectual history of DUE, as well as offer comparisons of DUE mathematical formulations of numerical algorithms. Those papers are the following: 1. Cascetta and Cantarella (1993). Modeling dynamics in transportation networks: state of the art and future developments. Presents a general framework for the simulation of dynamics in transportation networks. Models and algorithms for both within-day and day-to-day dynamic traffic assignment are discussed. The proposed framework includes static models as a particular case and is general enough to cover many existing models. 2. Peeta and Ziliaskopoulos (2001). Foundations of dynamic traffic assignment: the past, the present, and the future. Considers numerous DTA formulation and solution approaches, including those based on the formalisms of mathematical programming, variational inequalities, optimal control, and simulation. The aim is to document the main existing DTA approaches as of the date of publication, for future reference. Summarizes the current understanding of DTA, reviews the existing literature, and hypothesizes about the future of the discipline. 3. Boyce et al. (2001). Analytical models of the dynamic traffic assignment problem. Considers a restricted subset of analytical formulations of the dynamic traffic assignment problem, focusing on the authors’ experience with variational inequalities. Solution algorithms and computational issues requiring additional study are identified.

1.2 Some DTA and DUE Literature

3

4. Szeto and Lo (2005a). Dynamic traffic assignment: review and future research directions. Stresses the fidelity and accuracy of DTA models for offline network planning and policy evaluations, as well as real-time operation and management. Reviews analytical DTA formulations. Also compares DTA formulations in terms of their use of traffic flow theory and suggests future research directions. 5. Szeto and Lo (2005b). Properties of dynamic traffic assignment with physical queues. Narrowly focuses on the literature addressing physical queues, and advocates spillback models instead of point-queue concepts be used for DTA. Compares the properties of physical-queue DTA to point-queue DTA, emphasizing the first-in-first-out (FIFO) queue discipline and queue spillback, along with causality and travel-time-link-flow consistency. 6. Mun (2007) Traffic performance models for dynamic traffic assignment: an assessment of existing models. Provides a review of traffic performance models for dynamic traffic assignment (DTA), and identifies the strengths and weakness of existing models. Requirements for traffic performance models identified and various forms of existing traffic performance models for DTA reviewed and analyzed. Nonlinear travel time models shown to have deficiencies that may make them unsuitable for the analysis of time-varying transportation networks. Limitations of linear travel time models are discussed. Reconciliation of inconsistencies among travel time models is viewed as mandatory for theoretically coherent and plausible DTA modeling. 7. Jeihani (2007). A review of dynamic traffic assignment computer packages. Reviews the dynamic traffic assignment models contained within some wellknown computer packages. Describes demand estimation, supply representation, methods for computing dynamic user equilibria, and convergence. Concentrates mainly on TRANSIMS, which estimates second-by-second movements of individual travelers while exploiting parallel processing and cellular automata representations. TRANSIMS addresses some of the existing problems in dynamic traffic assignment models, although it cannot address existence, stability, uniqueness, and other important qualitative properties. 8. Rakha and Tawfik (2009). Traffic networks: dynamic traffic routing, assignment, and assessment. Describes dynamic travel behaviors and models thereof. Such behavioral models are argued to be essential to DTA modeling. Also conjectures what future models will be needed and their essential properties. 9. Viti and Tampère (2010). Dynamic traffic assignment: recent advances and new theories toward real-time applications and realistic travel behavior. Introduction to a volume of selected refereed papers from the DTA 2008 Symposium in Louven, June 18-20, 2008. 10. Szeto and Wong (2012). Dynamic traffic assignment: model classifications and recent advances in travel choice principles. Focuses on the travel choice principle and the classification of DTA models. Implications of the so-called travel choice principle for existence and uniqueness of DTA solutions discussed. Claims to explain the relationship between the travel choice principles and traf-

4

1 Introduction

fic flow using nonlinear complementarity, variational inequality, mathematical programming, and fixed-point model formulations. The above review papers consider hundreds of publications. Even so, a great deal of progress in formulation, mathematical analysis, and solution of DTA models has occurred since 2012, the date of the most recent review paper listed in Sect. 1.2. In the balance of Chap. 1, we describe papers that have been published since that date.

1.3 Vocabulary of DUE Modeling It is helpful to distinguish DUE papers from one another according to the following categories of model formulations, algorithms, analyses, and results: 1. Pure Traffic Assignment. A critical aspect of DUE modeling is the choice of assumptions employed to represent the route choice, departure-time choice, and other behaviors of agents acting on the traffic network of interest. The notion of pure traffic assignment assumes that pertinent delay operators are available from a separate model. As a consequence DSO and DUE formulations that are highly similar to their static SO and UE predecessors may be constructued, in either discrete or continuous time. 2. Network Loading. The dynamic network loading (DNL) problem, as we shall see subsequently, is generally a subset of the expressions used in the formulation of an analytical DUE model. In that case, it expresses how unit effective travel delay may be expressed in terms of departure rates and clock times. It should be noted, however, that most simulation-based DUE models contain subroutines that determine arc and path delays corresponding to specific present and prior traffic levels. Thus, it is possible for a DUE simulation model to be used to perform network loading; thereby DUE simulation models may be used in conjunction with pure traffic assignment models. Moreover, network loading may be based on a number of different assumptions, including Vickrey’s congestion model, the generalized Vickery model, the link delay model, and Lighthill-WhithamRichards hyrodynamic traffic flow theory. In Sect. 1.6.2, we provide citations and discussion of these and other foundations for the DNL problem. 3. Mathematical Formalism. We will use the phrase “choice of mathematical formalism” to mean the choice of mathematical perspective for and the representation of salient phenomena intrinsic to a given model of DTA. The choices of formalism include nonlinear mathematical program (NLP), variational inequality (VI), nonlinear complementarity problem (NCP), fixed-point problem (FPP), differential algebraic equation (DAE), and simulation. Formalism choice is a nontrivial matter. In particular, (i) algorithms are often tied to a specific formalism; (ii) a problem expressed via one formalism may be difficult or even impossible to express in another formalism; and (iii) problem representation using a formalism unfamiliar among the intended audience may lead to scientific misunderstanding.

1.3 Vocabulary of DUE Modeling

5

4. Traffic Physics. In traffic science, phenomena are modeled, to some extent, according to analogies based on natural phenomena, especially the following: kinematics, heat diffusion, and the kinetic theory of gasses. These perspectives have a role to play in DTA, especially during network loading. The LighthillWhitham-Richards (LWR) theory of kinematic waves is by far the most widely endorsed type of traffic physics. 5. Economics and Game Theory. In that the Wardropian user equilibrium perspective, for both static and dynamic traffic assignment, may be linked to a Nash game, the connection of DTA to differential noncooperative game theory is well established. Even so, only a relatively small fraction of the total body of knowledge known as “differential game theory” has been exploited to date in the study of DTA. Other notions from economics, especially the theory of travel demand and the theory of congestion pricing, have been successfully integrated within DTA models. However, the notion of learning in DTA games has been inadequately addressed, and evolutionary game theory (EGT), the body of work most closely related to learning, all but ignored. 6. Multiple Timescales. There are two fundamental timescales that are intrinsic to the study of dynamic traffic assignment. These are the day-to-day (or inter-day) timescale and the within-day (or intra-day) timescale. The terminology “day-today” is by convention used loosely for models relevant to any large chunk of time, such as the morning or evening commute. Generally speaking, the day-to-day perspective has been advocated for demand formation and for approximation of within-day phenomena when solution times must be shortened for near real-time traffic management. Most DUE models published to date are single timescale, within-day models. 7. Multiple Spatial Scales. It is self-evident that DTA involves phenomena whose representation depends on the spatial scale chosen. An example from analytical DUE and DSO modeling is the very nearly universal aggregation of individual vehicles to form origin-destination traffic volumes and so-called path flows. 8. Differential Algebraic Equations. It is helpful at this point to introduce the notion of a differential algebraic equation (DAE) system. In particular, a DAE system is a collection of equations involving state variables and their derivatives, and it is parameterized in terms of control variables. An ordinary differential algebraic (ODAE) system will have the form F (x, x, ˙ h) = 0

.

(1.1)

where x is a vector of state variables (traffic volumes) and h is a vector of departure rates for DNL models like the Vickrey-type models discussed in Sect. 1.6.2 below. It is also the case that network loading models involving hydrodynamic analogies may be expressed as partial differential algebraic equation (PDAE)

6

1 Introduction

systems of the general form  F

.

 ∂k ∂ρ , ,h = 0 ∂t ∂x

(1.2)

where concentration k and flow f are functions of location x and time .t from DNL models based on the Lighthill-Whitham-Richards kinematic wave equation discussed in Sect. 1.6.3, while h remains a vector of departure rates. It is important to recognized that DAE (PDAE) systems are not necessarily reducible to systems of ordinary (partial) differential equations. For our purposes in this book, it is enough to know that constrained differential equations are one type of differential algebraic equation system.

1.4 Alternative Formulations of DUE Numerous scholarly teams working independently around the globe have slowly made advances in modeling and computing dynamic use equilibria. In fact, DUE modeling and computation have now reached a point where substantial agreement exists regarding the general content of a mathematical model of dynamic user equilibrium, as well as the desired standards of performance for algorithms that compute DUE flow patterns. Friesz et al. (2011) digested the DTA and DUE literatures mentioned in Sect. 1.2 and found five broad groupings of modeling focus and associated unanswered research questions; these are: 1. Submodels and their simultaneous solution. We may distinguish two essential aspects of modeling dynamic user equilibrium for the within-day timescale: (i) dynamic network loading and (ii) simultaneous route and departure time equilibria, where dynamic network loading (DNL) subsumes the modeling of delay, flow evolution (arc dynamics), and flow propagation (enforcement of traffic laws during flow evolution). In its most general form, dynamic traffic assignment (DTA) determines time-varying departure rates, route choices, link volumes, and travel demands consistent with the physics of traffic and established travel demand theory. However, it is both useful and common to envision DTA as composed of two subproblems: (1) pure dynamic user equilibrium (PDUE) and (2) dynamic network loading (DNL). In PDUE we are concerned exclusively with the calculation of departure times, departure rates, and the routes (paths) employed. By its very nature PDUE requires some sort of behavioral model to describe the route and departure time choice process. In contrast, the focus of DNL is on the calculation of arc volumes, delays encountered, and queue spillbacks at intersections into upstream arcs, if any. It is important to recognize that expression of the DNL and PDUE subproblems does not imply that the decisions intrinsic to each subproblem are meant to be calculated sequentially. Quite the contrary, the DNL and PDUE subproblems should be

1.5 The Structure of DUE Models

2.

3.

4.

5.

7

solved simultaneously, and their solutions must be compatible in the sense that algorithms for their solution must yield routes, route delays, departure rates, and travel demands that are mutually consistent across subproblems. Complementarity and variational inequality formulations. Simultaneous route and departure time choice are integral to the definition of a dynamic user equilibrium and have to date been mainly expressed as variational inequalities, quasi-variational inequalities, or complementarity problems, either in discrete time or continuous time. However, the emerging literature on abstract differential variational inequalities has not been well exploited for either modeling or computing simultaneous route and departure time equilibria. Mathematical formulation of the network loading subproblem. Little agreement exists regarding an appropriate mathematical formulation of network loading. Furthermore, the emerging literature on differential algebraic equations, despite its focus on problem structures like those encountered in network loading, has not been exploited. Convergence without monotonicity. Fully general path delay operators may fail to be monotonic and/or differentiable. Rigorously convergent algorithms for determining path departure rates that constitute a user equilibrium for such general path delay operators are not presently available. We will discuss this circumstance in greater depth in subsequent chapters of this book. Multiple timescales. Among analytical DUE models, there are few multitimescale models recognizing tactical routing and departure time decisions are made in continuous time (the within-day timescale), while demand evolves in discrete time (the day-to-day timescale), and the two timescales are coupled, although there is considerable agreement that this dichotomy of timescales is apropos. In this “story,” the within-day timescale is based on continuous time, while the day-to-day timescale is based on discrete time.

1.5 The Structure of DUE Models Available DUE and DUE-related models tend to be composed of three to eight essential submodels drawn from this list: 1. 2. 3. 4. 5. 6. 7. 8. 9.

a model of effective path delay; fixed trip table or elastic travel demand model; path flow conservation constraints; constraints assuring non-negativity of path flows; a route and departure-time choice model; arc flow dynamics; arc flow propagation constraints; constraints assuring non-negativity of arc flows; and a day-to-day model of demand growth.

8

1 Introduction

Dynamic user equilibrium models from the early 1990s forward have been largely concerned with the so-called within-day timescale for which drivers make tactical routing and departure decisions. The notion of “day” here is quite arbitrary and could be any portion of an actual day for which there is a significant, discernible fluctuation in travel demand relative to some preceding or future “day.” As we have noted above, it has become commonplace to use the appellation dynamic network loading (DNL) to refer to the determination of arc-specific volumes, arc-specific exit rates, and experienced path delay when departure rates are known for each path. As such, DNL is typically represented by submodels 1, 6, 7, and 8 above. Moreover, when taken together, we refer to submodels 1, 2, 3, 4, and 5 as pure dynamic user equilibrium (PDUE). Note that both DNL and PDUE involve submodel 1; this is because DUE determines departure rates for given path delays and DNL determines path delays for given departure rates. Submodel 9, demand growth, occurs on the day-to-day timescale and allows travel demand to evolve. The DNL and PDUE problems, presented in subsequent sections, are sometimes presented as a single model, as in Friesz et al. (1993), Ran and Boyce (1996b) and Smith and Wisten (1995). In other works, as delineated below, the DNL problem is viewed as a mechanism that defines an implicit travel delay operator used in computing solutions of the DUE problem. In the latter case, however, there is no suggestion of a sequential decision process or a sequential numerical solution technique. Instead, the “operator approach” recognizes that the complete problem of interest has a form consistent with the following relationships: .

DUE: F (Ψ, x, h) = 0.

(1.3)

DNL: G (Ψ, x, h) = 0

(1.4)

where x is an abstract state vector, .Ψ is a vector of effective travel delays, and h is a vector of departure rates. Under appropriate regularity conditions, the DNL problem will determine the state vector .x = φ (t) and the delay operator .Ψ (t, h) when the control vector h is stipulated. As a consequence, the dynamic user equilibrium problem becomes F (Ψ (t, h) , h) = 0

.

(1.5)

Thus, the original problem of interest is reconceived as problem (1.5) that involves only departure rates, making it thereby a PDUE problem. Expressions (1.3), (1.4), and (1.5) are meant to capture the relationship of various DNL models and PDUE. The operators .F (·) and .G(·) are abstract representations of selected submodels drawn from the list above, according to the model builder’s creative vision. A much more delicate issue than choice of formalism is that of deciding how to deal with the network loading process for the formalism one chooses. That is to say, there are two perspectives on network loading within DUE models based on structures akin to (1.4); these are:

1.6 Dynamic Network Loading Models

9

1. The most obvious perspective for DUE model construction is to express the selected dynamic network loading (DNL) model as additional, explicit constraints of the DUE model (1.4). This perspective has been employed by Bernstein et al. (1993), Friesz et al. (2001), Lo and Szeto (2002), Szeto and Lo (2004), and Friesz and Mookherjee (2006). In addition, Chow (2009), when modeling dynamic system optimal (DSO) flows with departure time choice, presented a DNL model as explicit constraints of his DSO model. Generally speaking, the aforementioned contributions reduce their DUE models to so-called differential variational inequalities, a topic we discuss in depth in subsequent chapters. However, there are versions of DUE reminiscent of (1.3) with explicit DNL constraints that have been expressed as so-called mixed complementarity problems and differential complementarity systems by Ban et al. (2008), Ramadurai et al. (2010), and Pang et al. (2012). It is not presently known whether the models put forward by Ban et al. (2008) and Ban et al. (2012) are equivalent to the DUE formulations of Friesz et al. (1993), Friesz et al. (2001) and Friesz and Mookherjee (2006). 2. Another perspective for DUE model construction, and the one we presume herein whenever we distinguish the pure dynamic user equilibrium (PDUE) problem from the dynamic network loading (DNL) problem, is to view the DNL problem as a computable operator that specifies effective travel delay anytime the departure rates (path flows) are known, as presented in Xu et al. (1999), Friesz et al. (2011), Han et al. (2013c) and Han et al. (2015a,c). In this perspective, the effective delay operator may (but does not have to) be approximated as a nonlinear response surface, as in Song et al. (2017). Note that such a “delay operator approach” is neither a simulation nor a sequential modeling approach, as has sometimes been mistakenly asserted. In both of the perspectives listed immediately above, the calculated DUE will be the same; neither is the approximation of the other.

1.6 Dynamic Network Loading Models Generally speaking, dynamic network loading (DNL) models have all or a subset of the following components: 1. 2. 3. 4. 5.

some form of link and/or path dynamics; constraints assuring arc exit flows that do not exceed exit flow bounds; constraints assuring arc entrance flows do not exceed entrance flow bounds; constraints enforcing the fundamental diagram; a presumed state law expressing the relationship between traffic density and speed; 6. an equation describing the relationship between link traversal time and speed; 7. flow propagations constraints; and 8. appropriate initial conditions.

10

1 Introduction

In the remainder of this section, we discuss various perspectives for creating dynamic network loading (DNL) models.

1.6.1 Vickrey’s Model of a Traffic Bottleneck and Its Extension Vickrey (1963), Vickrey (1969) gave a mathematical description of congestion in a traffic bottleneck; his formulation may be considered a dynamic network loading (DNL) model. The essential insight provided by Vickrey is that there exists a tradeoff between travel time and schedule delay. Vickrey’s model of congestion was employed by Vickrey and Sharp (1968) to study the welfare effects of congestion pricing. Arnott et al. (1990) and Arnott et al. (1999) used Vickrey’s framework to further study the economics of bottlenecks in both deterministic and stochastic settings. More recent looks at congested bottle necks include the works by Yao et al. (2010) and Yao et al. (2012). Vickrey’s work served as the foundation of several early investigations of dynamic user equilibrium, including Arnott et al. (1992), who study a corridor with two parallel routes, and Iryo (2013), who reviews various schemes for extending the Vickrey bottleneck model to support dynamic user equilibrium analyses for more general network topologies. The aforementioned literature on the Vickery model employs two fundamental assumptions: (1) the vehicles of interest do not occupy any physical space, and, therefore, queues of vehicles likewise do not occupy space; and (2) link traversal time consists of a fixed travel time plus a queuing time that represents congestion. We choose to discuss a formulation of Vickrey’s model of a congestion bottleneck found in expositions by Kuwahara and Akamatsu (1997) and Nie and Zhang (2005). The relevant notation is ua (t) = the flow that enters the arc of interest a at time t

.

va (t) = the flow that exits arc a at time t qa (t) = the number of vehicles in queue at the exit node of arc a Ma = the capacity of the bottleneck expressed as flow (e.g., veh/sec) Ta = the free flow travel time for the arc Da (t) = the arc traversal time of vehicles entering arc a at time t We consider the time window [t0 , tf ], a segment of the real line for which tf > t0 . Vehicles entering the link move at their free flow speed before arriving at the exit, which is the aforementioned bottleneck. Moreover, a point queue forms at the exit node of the arc of interest. The rate at which vehicles are released from the queue, va (t), is described as follows: vehicles are released at the maximal rate allowed by the arc’s physical capacity and vehicles who have already arrived to form the queue; the discharge

1.6 Dynamic Network Loading Models

11

rate from the link (and the queue) is min {ua (t − Ta ), Ma }. These dynamics are conveniently expressed as .

dqa (t) = dt



0 if t0 ≤ t < Ta ua (t − Ta ) − va (t) if Ta ≤ t ≤ tf

where va (t) = min{ua (t − Ta ), Ma }. Some additional thought reveals that Da (t) = Ta +

.

qa (t + Ta ) Ma

for t ∈ [t0 , tf ]; therefore .

 dqa (t) min {ua (t − Ta ), Ma } if qa (t) = 0 = ua (t − Ta ) − if qa (t) > 0 Ma dt

(1.6)

may be used as the dynamics of the Vickrey model, although one must be careful to note that (1.6), as an ordinary differential equation, contains a discontinuity. In an effort to develop a formulation and solution procedure of the complete DUE problem (DNL as well as PDUE) Pang et al. (2012) recast the Vickrey bottleneck model as a differential linear complementarity system. Their result, although interesting, does not carry with it any new physical interpretation or enhanced computational efficiency. Another close relative to the Vickrey model is that due to Ban et al. (2012), who transform (1.6) into an explicit ordinary differential equation (with continuous right-hand side) by introducing a parameter α so that the Vickrey dynamics are approximated as: .

dqa (t) = min {ua (t − Ta ) − Ma , −αqa (t)} dt

(1.7)

In (1.7) Ban et al. (2012) assume α >> 1, and, as a consequence, refer to it as the α-model. Although (Ban et al., 2012) are able to show the equivalence of (1.7) to the Vickrey model in the limit α −→ +∞, the α-model does not constitute a significant generalization of Vickrey’s original work. Moreover, it is now widely felt that generalization of Vickrey-like models for DNL does not hold great promise for treating queue spillback because of the nonphysical queuing perspective at the heart of Vickrey’s original model. In a subsequent paper, Ban et al. (2012) have shown that Vickrey’s model may be expressed as a differential complementarity system for which it is asserted that implicit time discretization provides superior computational performance. Several points are in order: (1) without a complete explanation of the notion of a differential complementarity system, the model reformulation found in Ban et al. (2012) cannot be fully appreciated; (2) the claim of superior computational performance is based on relatively few numerical tests; (3) indeed, as we will show, other numerical

12

1 Introduction

approaches also hold great promise for solving DNL models; and (4) Vickrey’s framework is not a springboard to deeper and more behaviorally well-founded DNL models.

1.6.2 Network Loading Based on Link Dynamics Analytical DUE models developed in the early 1990s were influenced greatly by the dynamic system optimal model of Merchant and Nemhauser (1978a,b) who proposed an especially simple type of arc dynamics that still influences presentday DUE models. In particular, if one posits that it is possible to specify, or to mathematically derive from some plausible theory, functions that describe the rate at which traffic exits a given network arc for any given volume of traffic present on that arc, one is lead to some deceptively simple arc dynamics. To express this supposition symbolically, let .xa (t) denote the volume of traffic on arc a at time .t and let .ga [xa (t)] be an exit function that gives the rate at which traffic exits from link a. Also let the rate at which traffic enters arc a be denoted by .ua (t). Note that both .ga [xa (t)] and .ua (t) are rates; that is, they have the units of volume per unit time, so it is appropriate to refer to them as exit flow and entrance flow, respectively. A natural flow balance equation can now be written for each link: .

dxa = ua (t) − ga [xa (t)] dt

∀a ∈ A

(1.8)

where every arc of the network of interest is directed and .A denotes the set of all arcs. In Merchant and Nemhauser (1978a,b) each .ua (t) is treated as a control variable. Although (1.8) may seem an obvious identity, it is actually an approximation that only becomes exact when arcs are of infinitesimal length and traffic interactions with other links do not occur. The same dynamics were employed by Friesz et al. (1989) and Wie et al. (1995) in an effort to develop a model of dynamic user equilibrium. However, their model relies on dynamic shadow prices that are difficult to interpret and does not yield true dynamic user equilibria relative to route and departure time decisions. Moreover, the exit flow functions they use in conjunction with (1.8) may be criticized as difficult to specify and measure, and they may produce violations of the first-in-first-out queue discipline. Exit flow functions allow for the potential violation of the first-in-first-out (FIFO) queue discipline as illustrated and discussed by Carey (1986), Carey (1987), and Carey (1992). However, Carey and McCartney (2004) show that the violation of FIFO is largely dependent upon time and space discretization schemes. Additionally, Carey (2004a,b) show that exit flow functions may satisfy FIFO but not causality. Another problem with exit flow functions is that an inflow at any time t affects the outflows at the same time instant t. These difficulties have caused almost all researchers studying dynamic network flow problems to abandon dynamics based on exit flow functions. We also point out that the Merchant-Nemhauser model employed flow

1.6 Dynamic Network Loading Models

13

conservation constraints of the following form for the case of a single origindestination pair: Sk (t) =



ua (t) −

.

a∈A(k)



ga [xa (t)]

∀k ∈ N

(1.9)

b∈B(k)

where .A (k) is the set of arcs with tail node k, .B (k) is the set of arcs with head node k, and .N is the set of all network nodes. An intriguing modification of the Merchant-Nemhauser arc dynamics was proposed by Ran et al. (1993). Their idea was to employ dynamics that treat both arc entrance and exit flows as control variables; that is, ij

.

dxa ij ij = ua (t) − va (t) dt

∀a ∈ A, (i, j ) ∈ W

(1.10)

where .P is the set of all network paths and .W is the set of all origin-destination ij pairs, .xa (t) is the flow on arc a traveling between origin-destination pair .(i, j ) ∈ ij ij W, while .ua (t) and .va (t) denote the rates at which traffic also traveling between .(i, j ) enters and exits arc a, respectively. Two types of flow propagation constraints for preventing instantaneous flow propagation and ensuring the FIFO queue discipline have been suggested by Ran et al. (1993) and Ran and Boyce (1996b) for dynamics (1.10). The first is stated as: p

p

Ua (t) = Va [t + Δa (t)]

.

p

p

∀a ∈ A, p ∈ P

where .Ua (·) and .Va (·) are the cumulative numbers of vehicles associated with path p, which are entering and leaving link a, respectively, while .Δa (t) denotes the time needed to traverse link a at time t and .P is the set of all paths. The meaning of these constraints is fairly intuitive: vehicles entering an arc at a given moment in time must exit at a later time consistent with the arc traversal time. The second Ran et al. type of flow propagation constraint is much more notationally complex and is omitted here for the sake of brevity. Suffice it to say that constraints of this second type are articulated in terms of path-specific arc volumes and are meant to express the idea that path-specific traffic on an arc must ultimately visit a downstream arc or exit the network at the destination node of the path in question. Ran et al. argue that by enforcing this consideration, they rule out FIFO violations and instantaneous flow propagation anomalies. Ran et al. (1993) employ a Beckmann-type objective function to create an optimal control model with dynamic user equilibria as solutions. Bernstein et al. (1993) introduced the notion of exit time functions together with a variational inequality to describe dynamic user equilibrium. Their exit time function notion is consistent with FIFO for appropriate arc delay functions. Following Friesz et al. (1993) we fix a network .G(A, V) expressed as a directed graph with .A being the set of links and .V being the set of nodes. Let .P be the set of paths employed by

14

1 Introduction

travelers, and .W be the set of origin-destination pairs. Each path .p ∈ P is expressed as an ordered set of links it traverses: p = {a1 , a2 , . . . , am(p) }

.

where .ai denotes the i-th link, and .m(p) is the number of links in path p. In particular, Bernstein et al. (1993), Friesz et al. (2001) and Friesz and Mookherjee p (2006) use a function .ξai (t) that expresses the time of exit from arc .ai of every path   p = a1 , a2 , . . . , ai−1 , ai , ai+1 , . . . , am(p) ∈ P ,

.

The exit time functions obey the recursive relationships  p ξa1 = t + Da1 xa1 (t)

.

 p  p p ξai = ξai−1 (t) + Dai xai ξai−1 (t)

.

∀p∈P

∀ p ∈ P, i ∈ {2, . . . , m(p)}

(1.11) (1.12)

 where .Dai xai (t) is the time to traverse arc .ai ; it is a function of the number of vehicles .xai in front of the entering vehicle at the time of entry. This model of arc delay is herein called the link delay model (LDM). Friesz et al. (1993) also give a continuous time articulation of flow conservation based on a fixed within-day trip matrix: 

tf

.

p∈Pij

hp (t) dt = Qij

∀ (i, j ) ∈ W

(1.13)

t0

where .Pij is the set of paths connecting .(i, j ) ∈ W and .hp (t) is the departure rate from the origin  of path .p1 ∈ Pij , while .Qij is the fixed travel demand between ∈ R+ is the continuous time interval representing a single .(i, j ) ∈ W and . t0 , tf day or commuting period of interest. They used (1.11) and (1.12) together with dynamics expressed as integral equations involving inverse exit time functions to define an effective path delay operator. That operator, in turn, was used with (1.13) and non-negativity restrictions to construct an infinite dimensional variational inequality whose solutions are dynamic user equilibria. Moreover, Friesz et al. (1993) provided the first expression of dynamic user equilibrium as a variational inequality. Subsequently, Wu et al. (1998) and Xu et al. (1999) developed algorithms for the DNL submodel considered by Friesz et al. (1993). In particular they studied the use of the projected gradient method and solved some modest size test problems, but did not provide useful convergence results. It seems likely that Wu et al. (1998) were the first to employ the name dynamic network loading for the type of model we are considering in this section.

1.6 Dynamic Network Loading Models

15

We reiterate that Friesz et al. (2001) employed path delays computed from (1.11) and (1.12) with dynamics p

.

dxa1 (t) p = hp (t) − ga1 (t) dt

∀p ∈ P.

(1.14)

p

dxai (t) p p = gai−1 (t) − gai (t) dt

∀ p ∈ P, i ∈ {2, . . . , m(p)} ,

(1.15)

p

where .xai (t) is the volume of traffic on arc .ai of path p for .i ∈ {1, . . . , m(p)} p and .gai (t) denotes the flow exiting that same arc, to formulate the dynamic user equilibrium problem as a differential variational inequality that is completely equivalent to the Friesz et al. (1993) infinite dimensional variational inequality formulation. Friesz et al. (2001) included in their formulation the flow propagation constraints

  

 ga1 t + Da1 xa1 (t) 1 + Da 1 xa1 (t) x˙ai (t) = hp (t)

.

(1.16)

  

 p

p gai t + Dai xai (t) 1 + Da i xai (t) x˙ai (t) = gai−1 (t)

.

.

∀ p ∈ P , i ∈ {2, . . . , m(p)}

(1.17)

which are identical to those proposed by Astarita (1995) and which include consideration of expanding/contracting platoons of vehicles. Friesz et al. (2001), Friesz and Mookherjee (2006), and Friesz et al. (2011) look at network loading from the perspective of (1.11), (1.12), (1.14), (1.15), (1.16), and (1.17). The paper by Li et al. (2000) is one of several that uses the Friesz et al. (1993) recursive equations (1.11) and (1.12) that are based on exit time functions along with the flow propagation constraints (1.16) and (1.17) to express path delay and assure physically meaningful flow. Friesz et al. (2001) show that the network loading mechanism described immediately above may be used to form a DUE model whose solutions are dynamic user equilibria. They offer an ad hoc algorithm without discussing convergence. The recursive equations (1.11) and (1.12) and flow propagation constraints (1.16) and (1.17) also have much in common with Tong and Wong (2000) in their study of dynamic user equilibrium with bounds on queue length. Huang and Lam (2002) determine path delay using a nested delay function reminiscent of that arising from the recursive relationships (1.11) and (1.12). Friesz et al. (1993), Friesz et al. (2001), and Friesz and Mookherjee (2006) have employed network models that are based on the notion of a link delay function expressing delay as a function of traffic in front of a user upon entering a given arc. A generic DNL model using the link delay perspective across an entire network relies on  1 if ai ∈ p .δai p = /p 0 if ai ∈

16

1 Introduction

to be an element of the arc-path incidence matrix. Of course total path traversal time is Dp (t) =

m(p) 

 p p p ξai (t) − ξai −1 (t) = ξam(p) (t) − t ∀ (i, j ) ∈ W, p ∈ Pij

.

i=1

since we use the convention that p

ξ0 (t) = t ∀ (i, j ) ∈ W, p ∈ Pij

.

Next we note that arc exit times depend on arc delay functions, while arc delays depend on arc volumes, per expressions (1.11) and (1.12). By virtue of expressions (1.14) through (1.17), we can see that knowledge of h completely determines individual arc volumes as well as all arc exit times and hence path traversal time. So one may express the forgoing apparatus for delay on path .p ∈ P as the operator .Dp (t, h). It follows that the so-called effective path delay is the operator  Ψp (t, h) = Dp (t, h) + Φp t + Dp (t, h) − A ∀p ∈ P

.

where A is the desired arrival time and .Φp [.] is the arrival penalty. The result of the foregoing discussion is the following .∀p ∈ P, i {1, . . . , m(p)}:

∈

p

.

dxai (t) p p = gai−1 (t) − gai (t) . dt  p dyij = gam(p) ∀ (i, j ) ∈ W. dt

(1.18) (1.19)

p∈Pij

 Ψp (t) = Dp (t, h) + Φp t + Dp (t, h) − A .

(1.20)

g ∈ Ω.

(1.21)

x (t0 ) = x 0.

(1.22)

y (t0 ) = 0

 y tf = Q

.

and

p  x p = xai : i = 1, . . . , m (p)

 x = xp : p ∈ P

p  g p = gai : i = 1, . . . , m (p)

.

(1.23) (1.24)

1.6 Dynamic Network Loading Models

17

 g = gp : p ∈ P 

y = yij : (i, j ) ∈ W while p

hp = ga0 ∀p ∈ P, i ∈ {1, . . . , m(p)}

 h = hp : p ∈ P

.

and the set .Ω is defined as Ω = {g ≥ 0 : (1.16), (1.17)}

.

p

The above makes clear that the link volumes .xai are natural state variables, while p the link entrance (exit) flows .gai are natural control variables. Note that system (1.18)–(1.24) must be well defined whenever the path flow (departure rate) vector is known. Said differently, system (1.18)–(1.24) must have a solution whenever path flow .h (t) is specified. We note that (Bliemer and Bovy, 2003) develop a DNL model that extends the arc delay model presented above to a formulation involving multiple user classes.

1.6.3 Network Loading as a LWR-Based PDAE System We now discuss network loading from the point of view of the Lighthill-WhithamRichards (LWR) hydrodynamic theory of traffic flow (Lighthill & Whitham, 1955; Richards, 1956). The LWR model in its most general form is a partial differential equation (PDE) together with appropriate boundary conditions. Multiple ways of invoking LWR dynamics to perform DNL have been presented in the DTA literature. The aim of this section is to formulate the LWR-based dynamic network loading (DNL) problem as a system of partial differential algebraic equations (PDAEs). The proposed PDAE system uses vehicle densities as the primary unknown variables and computes link dynamics, flow propagation, and path delay for any given set of departure rates along utilized paths. The PDAE system captures vehicle spillback explicitly and accommodates a wide range of junction types and so-called Riemann solvers.

1.6.3.1

The Perakis-Kachani DNL Models

In a series of papers Kachani and Perakis (2001), Kachani and Perakis (2002), and Kachani and Perakis (2009) provide very accessible discussions of how alternative LWR-based DNL models may be constructed; these works do not rely on highly technical and subtle properties of the LWR equation itself. Actually

18

1 Introduction

their formulations include not only DNL considerations, but also the conditions for dynamic user equilibrium as well. As we are presently interested in DNL models, we now only refer to the network loading aspects of their work. As in Sect. 1.6.2, we consider a network .G(A, V) expressed as a directed graph with .A being the set of links and .V being the set of nodes. Again let .P be the set of paths employed by travelers and .W be the set of origin-destination pairs. Each path .p ∈ P is expressed as an ordered set of links it traverses: p = {a1 , a2 , . . . , am(p) }

.

where .ai denotes the i-th link in path p and .m(p) is the number of links in this path. There are several crucial components of a complete LWR-based network loading procedure, each of which will be elaborated in a subsection below. Throughout this section, for each node .v ∈ V, we denote by .I v the set of incoming links and .Ov the set of outgoing links. We need to augment the notation introduced immediately above in order to illustrate the Kachani-Perakis style of DNL modeling. To that end consider the following: .|P |

.=

.Lp

.=

t)

.=

.|A|

.=

x .La .fa (x, t) in .Ca (t) out .Ca (t) .Ti (x0 , xa , t)

.=

.Ta (La ,

t) t) .ρa (x, t) .ρ max .ua max .fa .(a, p) .δap .Lap .Tap (Lap , t)

.=

.va (x,

.=

.[t0 , tf ]

.=

.Tp (Lp ,

.= .= .= .= .=

.= .= .= .= .= .= .= .=

cardinality of the set of paths .P length of path .p ∈ P unit path travel time cardinality of the set of arcs .A position on arc .a ∈ A the length of arc .a ∈ A the flow on arc .a ∈ A at position x at time t inflow capacity for arc a at time t outflow capacity for arc a at time t travel time for a driver at position .x0 of arc a at time t to reach position .xa traversal time for driver departing on arc a at time t travel speed on arc a at position x at time t traffic density on arc a at position x at time t vector of traffic densities on all arcs maximum traffic speed on arc a maximum storage of arc a arc-path pair 1 if a belongs to path p, 0 otherwise distance from the origin of path p to the entrance of arc a travel time from the origin of path p to the entrance of arc a when departing at time t the time interval of interest

1.6 Dynamic Network Loading Models

19

Kachani and Perakis (2009) impose the following assumptions in order to present their intended PDAE: 1. links in the network have no intermediary exits; 2. velocity .va for arc .a ∈ A can be expressed as a function .Fa depending only on the vector of density functions, i.e. .va = Fa (ρ, ∇ρ); and 3. for all .t ∈ [t0 , tf ], the arc exit flow rate .fa (La , t + τa ) may be approximated by a piecewise continuously differentiable function of .τa , known as .gat (τa ) at instant t. With the above considerations in mind, one of the Kachani-Perakis PDAE models is the following: .

∂ ∂ ρa (x, t) + fa (xa , t) = 0 ∂t ∂x

∀a ∈ A.   fa (La , t + Ta ) = min gat (Ta ) , Caout (t) ∀a ∈ A fa (0, t) ≤ Cain (t) ∀a ∈ A. fa (x, t) = va (x, t) ρa (x, t) va = Fa (ρ, ∇ρ) 1 dTa (x, t) = dx ua



(1.25) (1.26) . (1.27)

∀a ∈ A.

∀a ∈ A.

∀a ∈ A.

Ta (0, t) = 0 ∀a ∈ A. 



  Ta La , t + Tap Lap , t δap = Tp Lp , t ∀p ∈ P

(1.28) (1.29) (1.30) (1.31) (1.32)

a∈A

Models (1.25)–(1.32) are but one LWR-based PDAE model of dynamic network loading. Indeed, Kachani and Perakis (2009) present three such models.

1.6.3.2

The Han-Friesz Within-Link Dynamics for DNL

Let us now consider a perspective on dynamic network loading (DNL) that is based on a network whose arcs behave according to LWR theory subject to carefully articulated boundary conditions at the entrance and exit of each arc. In effect such models build an LWR PDAE system from an explicit disaggregation perspective rather than the Kachani-Perakis implicit disaggregation perspective discussed in Sect. 1.6.3.1.

20

1 Introduction

To begin, we consider arc dynamics, for each .a ∈ A, expressed via the scalar conservation law .

 ∂ ∂  ρa (t, x) + ρa (t, x) · va ρa (t, x) = 0 ∂t ∂x

(t, x) ∈ [0, T ] × [0, La ] (1.33)

subject to appropriate initial and boundary conditions. In (1.33), .ρa (t, x) and va (ρa (t, x)) represent the vehicle andspeed on arc a, respectively. Their

 . density product, .ρa (t, x) · va ρa (t, x) = fa ρa (t, x) is the flow. In order to explicitly incorporate drivers’ route choices, for every .p ∈ P such that p .a ∈ p one must introduce the function .μa (t, x) , .(t, x) ∈ [0, T ] × [0, La ], which represents, per unit of flow .fa (ρa (t, x)), the fraction associated with path p. We will subsequently call these variables path disaggregation variables (PDVs). For each moving car, the surrounding traffic flow is composed of path-specific contributions. As a car moves, the composition of surrounding traffic will not change since its surrounding traffic moves at the same speed under assumed first-in-first-out (FIFO) queue discipline when there is no overtaking. In mathematical terms, this means p there are constant path disaggregation variables, .μa (·, ·), along the trajectories of cars .(t, x(t)) in the corresponding space-time diagram, where .x(·) is the trajectory of a moving car on arc a. That is, .

.

 d p

μa t, x(t) = 0 dt

∀p such that a ∈ p,

which, according to the chain rule, becomes .

 ∂ p

d ∂ p μa t, x(t) + μa (t, x) · x(t) = 0, ∂t ∂x dt

which leads to another set of partial differential equations on arc a: .

 ∂ p ∂ p μa (t, x) + va ρa (t, x) · μa (t, x) = 0 ∂t ∂x

∀p such that Ii ∈ p,

(1.34)

where .ρa (t, x) is the solution of (1.33) . The following obvious identity holds  .

p

μa (t, x) = 1

whenever ρa (t, x) > 0

(1.35)

p a

where .p a means “path p contains (or traverses) arc a” and the summation appearing in (1.35) is with respect to p. By convention, if .ρa (t, x) = 0, then p .μa (t, x) = 0 for all p. In subsequent chapters of this book (Sect. 8.1.3), we show how the above arc dynamics may be employed to calculate delay operators for each path of interest, by incorporating realistic junction dynamics concerning both p .ρa (t, x) and .μa (t, x).

1.6 Dynamic Network Loading Models

21

1.6.4 Network Loading Based on the CTM This dynamic network loading model is based on a joint discretization of time and space of the LWR model, which is referred to as the cell transmission model (CTM), a name coined by Daganzo (1994). Relatively short segments of the road of interest are created and a discrete time index is used to specify the following constrained dynamics: nj (t + 1) − nj (t) = yj (t) − yj +1 (t) .    yj (t) = min nj −1 (t) , Cj (t) , α Nj (t) − nj (t)

.

(1.36) (1.37)

where t is now a discrete time index and a unit time step is employed. In the above, the subscript j refers to a spatially discrete physical “cell” of the highway segment of interest while .(j − 1) refers to the cell upstream. We take .nj (t) to refer to the traffic volume of cell j . Furthermore, .yj (t) is the actual flow from cell .j − 1 to cell j , while .Cj is the maximal discharge from cell j in a single time step, .Nj is the holding capacity of cell j , and .α is a parameter. Daganzo (1995) shows how (1.36) and (1.37) can be extended to deal with network structures through straightforward bookkeeping. The language introduced previously is readily applicable to the cell transmission model; in particular (1.36) are arc (cell) dynamics (although now several dummy arcs can make up a real physical arc) and (1.37) are flow propagation constraints. The cell transmission model also includes an implicit notion of arc delay. That notion, however, is somewhat subtle: namely, delay is that which occurs from traffic flowing in accordance with the fundamental diagram of road traffic. This is because (1.37), as explained by Daganzo (1994), is really a piecewise linear approximation of the fundamental diagram of road traffic. Furthermore, Lo and Szeto (2002) employed the CTM to create link dynamics; they also present a route travel time extraction procedure that allows the CTM to subsume the role of the effective path delay operator as that notion was introduced in Sect. 1.5 above. The (Lo and Szeto, 2002) model is expressed in discrete time and is, hence, finite dimensional. They use a discrete version of (1.13) to express flow conservation and describe the dynamic user equilibrium itself as a finite dimensional variational inequality that is the discrete time equivalent of the infinite dimensional variational inequality formulation of Friesz et al. (1993). They propose and test an alternating direction method for solving their variational inequality formulation. Convergence is proven based on the assumption that the relevant operators are cocoercive. Mappings that are monotone and Lipschitz continuous over their domain of definition are co-coercive, although the converse does not hold in general. As a consequence, the Lo and Szeto (2002) convergence result does not consider the potentially non-monotonic delay operators we address in subsequent sections of this book (Chap. 6). Szeto and Lo (2004) extend the Lo and Szeto (2002) modeling framework to consider dynamic user equilibrium with elastic travel demand. Nie and Zhang (2010) use the cumulative departure and arrival curves for each arc and each path to calculate travel times. They also use (1.13) to express

22

1 Introduction

flow conservation. To these model features, they add arc dynamics and nodal dynamics, the latter to enable the direct consideration of queues. Three different models of arc dynamics are considered: the point queue model, the spatial queue model, and the kinematic wave model based on the LWR theory of traffic flow. Nie and Zhang (2008) also present a rather general model of traffic flow through nodes, based on notions of virtual demand and virtual supply. The dynamic origindestination demand estimation problem itself is modeled as a finite dimensional variational inequality that is the discrete-time equivalent of the infinite dimensional variational inequality formulation of Friesz et al. (1993). Furthermore, Nie and Zhang (2008) propose to solve their variational inequality formulation using an equivalent mathematical program based on the idea of a gap function; they test feasible direction methods and the method of successive averages for solving that program. They too do not provide regularity conditions that assure convergence of the algorithms tested. We also note that (Ban et al., 2008) employ the arc dynamics (1.14) and (1.15) and flow propagation constraints that are a modified version of (1.16) and (1.17). They also express flow conservation at the nodal level using constraints similar to (1.9); as a consequence their definition of dynamic user equilibrium identifies least travel cost as the difference of nodal dual variables.

1.6.5 Network Loading Based on the Variational Approach In Chap. 8 of this book, we present a continuous-time network loading procedure based on the Lighthill-Whitham-Richards model (Lighthill & Whitham, 1955; Richards, 1956). This is done through a variational approach known as the LaxHopf formula, an introduction to which can be found in Sect. 2.8.2 of this book. 1 A system of differential algebraic equations (DAEs) is proposed for describing traffic flow propagation, travel delay, and route choices. This approach allows construction of an efficient computational scheme for large-scale networks. Friesz et al. (2013) embed this network loading procedure into the dynamic user equilibrium (DUE) model proposed by Friesz et al. (1993). The (Friesz et al., 2013) DUE model is solved as a differential variational inequality (DVI) using a fixed-point algorithm. Several numerical examples of DUE on networks of varying sizes are presented, including the Sioux Falls network with a significant number of paths and origindestination pairs (OD). The DUE literature has seen a widespread application of the variational principle in dynamic network loading, with varying contexts and vocabulary. The seminal work of Newell (Newell, 1993a,b,c) is among the earliest to apply variational theory to highway traffic modeling. The variational perspective was examined in depth by Daganzo (2005), Daganzo (2006), and Claudel and Bayen (2010,b) in the

1 The Lax-Hopf formula and modifications of it are collectively called the variational approach herein because they have an embedded optimization problem.

1.6 Dynamic Network Loading Models

23

context of data fusion and traffic state estimation. Yperman et al. (2005) propose the link transmission model (LTM) based on Newell’s variational theory as a discretetime approximation of the LWR model. The dynamics of the LTM are expressed in terms of variables associated with the entrance and exit of the link of interest and propagate traffic flows within a link implicitly via the variational principle. Similar to the CTM, the LTM is capable of capturing realistic phenomena such as shock waves (although implicitly) and vehicle spillback, and the latter enjoys greater computational efficiency. The LTM was later studied in a continuous-time modeling framework by Osorio et al. (2011), Jin (2015) and Han et al. (2016).

1.6.6 LWR Network Loading Based on Closed-Form Operators Taking a somewhat different approach, relying on physical length, Perakis and Roels (2006) derive path and arc travel delay functions for specific flow, spatial, and time regimes associated with the propagation of kinematic waves along a roadway. They do not consider traffic interactions among arcs incident to the same node, although their delay functions do include exogenous time-dependent functions associated with arc entry and exit that can in principle be adjusted to account for such interactions. They are also able to show that their derived delay functions are monotonic and continuous and preserve the FIFO queue discipline provided the rate of evolution of density obeys certain inequalities. Their models also require that spillbacks be limited to the immediate upstream arc when shock waves arise. Driven by the need to study analytical properties of DUE such as existence, uniqueness, and convergent solution algorithms, Song et al. (2017) take a fresh look at the DNL subproblem through statistical metamodeling. Specifically, they employ Kriging (Matheron, 1963) to derive a surrogate path delay operator, as an approximation of the LWR-based DNL model, although the metamodeling approach can be applied to virtually any DNL model. Not only is the resulting surrogate delay operator continuous and smooth, it is also available in closed form, allowing the DNL sub-model to be executed with far greater computational efficiency.

1.6.7 Dynamic User Equilibrium in Continuous Time Pure dynamic user equilibrium (DUE) assumes that a dynamic network loading operator is available, although there is no need for the DNL model to have closedform solutions, as we shall see in subsequent chapters. There are a few different ways a DUE may be formulated. Although some formulations are equivalent, it must be emphasized that not all are equivalent. Most published mathematical formulations express an open-loop version of a dynamic notion of user equilibrium based on some type of generalization of Wardrop’s so-called first principle. Some PDUE models treat both route choice and departure time choice as fundamental

24

1 Introduction

decisions, while others are concerned with either departure time choice or route choice, but not both. In this book, we are concerned only with simultaneous-routeand-departure-time choice, which we abbreviate as SRDT choice. Recall that we denote the effective delay in the presence of traffic pattern h by .Ψp (t, h). We also take .vij to be the notional least travel unit cost for all .(i, j ) ∈ W. In fact, in a rigorous measure-theoretic framework, .vij is defined to be the essential infimum of the effective path delay functions associated with the same O-D pair .(i, j ); see Sect. 3.1 for further details. As a consequence of such notation, the dynamic user equilibrium model for finding path departure rates .h∗ ≥ 0 is the following: ⎫ ∀(i, j ) ∈ W ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ∗ ⎪ vij = essinf Ψp (t, h ) : p ∈ Pij ⎪ ⎪ ⎪ ⎬  tf ⎪ ⎪ h∗p (t)dt = Qij ∀ (i, j ) ∈ W ⎪ ⎪ ⎪ t 0 ⎪ p∈Pij ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ∗ h ≥0

h∗p (t) > 0, p ∈ Pij ⇒ Ψp (t, h∗ ) = vij

.

(1.38)

Furthermore, the notation .Ψp (t, h∗ ) expresses the fact that the effective path delay is in effect a solution of the dynamic network loading model one selects to describe  a network of interest. Moreover, .Q = Qij : (i, j ) ∈ W is an exogenous, fixed trip table;2 and integration is in the sense of Lebesgue. Formulation (1.38) is one version of the continuous-time model presented in Friesz et al. (1993). Ran et al. (1993), Ran and Boyce (1996a,b), and Ran et al. (1996) also worked in continuous time and created a family of DUE models sharing a common foundation based on an equivalent optimal control formulation, for which they show the best responses of generic travelers constitutes a DUE flow pattern. Instead of viewing unit effective travel delay as an operator, they embedded the notion of network loading within their DUE model. They relied on highly specific arc entrance and exit flow functions, as well as novel flow propagation constraints that are quite different than the continuous time physical identities employed by Friesz et al. (2001), Perakis and Roels (2006), Kachani and Perakis (2006), and Kachani and Perakis (2009) whose work is discussed in Sect. 1.6.2. A notion different from the SRDT DUE is the so-called route-choice (RC) DUE, which uses the following alternative flow conservation constraints in lieu of

2 We may accommodate elastic travel demand, as shown subsequently in later chapters of this book, without any great effort or complication.

1.6 Dynamic Network Loading Models



.

p∈Pij

tf

t0

25

h∗p dt = Qij for all .(i, j ) ∈ W:  .

hp (t) = Qij (t)

∀ (i, j ) ∈ W

(1.39)

p∈Pij

where .Qij (t) is the demand rate (vehicles per unit time) between O-D pair .(i, j ). The relaxation of (1.13) represented by (1.39) comes at the expense of behavioral richness; in particular, (1.13) allows drivers to depart in platoons as is generally observed in real networks; and (1.39) eliminates platoon formation. Friesz and Meimand (2014) and Han et al. (2015a) extend formulation (1.38) to treat elastic travel demand, while (Han et al., 2013c) provide a comprehensive existence theory. Moreover, working in continuous time, Bliemer and Bovy (2003) have extended the (Friesz et al., 1993) formulation by introducing multiple user classes, thereby creating a quasi-variational inequality. More recently, Ban et al. (2012) have introduced a continuous-time model of the so-called instantaneous dynamic user equilibrium that considers departure time choice but not simultaneous departure time and route choice. Moreover, the (Ban et al., 2012) model employs a Vickrey-type DNL sub-model. As we have commented previously, Vickrey-type network loading brings with it limitations on behavior and does not reside at the current DNL research frontier. The (Ban et al., 2012) instantaneous perspective ignores the critical notion of route choice, and, as such, its solutions arguably do not constitute a user equilibrium. In fact, the (Ban et al., 2012) instantaneous perspective has more in common with the early DTA literature on timing departure decisions (such as Friesz et al. 1989) than it does with the widely adopted notion of SRDT DUE.

1.6.8 Dynamic User Equilibrium in Discrete Time Although they were concerned solely with dynamic system optimal traffic flow, Merchant and Nemhauser (1978a,b) set the stage for future work on DUE by using a discrete-time dynamic programming approach. Subsequent discrete-time DUE models include those by Drissi-Kaïtouni and Hameda-Benchekroun (1992), Wie et al. (1995), Huang and Lam (2002), and Nahapetyan and Lawphongpanich (2007). More recently Lo and Szeto (2002) and Lo and Szeto (2004) have also made significant contributions to discrete-time DUE modeling with embedded CTM to accomplish DNL. In particular, their work made clear that LWR-based DNL could be integrated with pure dynamic user equilibrium and, thereby, launched many investigations into LWR-based DNL/DUE. The authors cited above for their work on discrete-time DUE used a variety of formulations including mixed integer linear programming, nonlinear programming, dynamic programming, optimal control theory, complementarity, and variational inequalities; in some instances,

26

1 Introduction

they were merely concerned with departure time choice and in other instances with simultaneous route and departure time choice.

1.7 Other Considerations in Classifying DUE Models It has been recognized for some time that so-called bounded rationality has an important role to play in modeling dynamic user equilibria (Mahmassani & Chang, 1987). More recently Guo and Liu (2011), Wu et al. (2013), and Han et al. (2015c) have analyzed bounded rationality for an array of timescales and notions of dynamic adjustment processes. As such the presence and style of expressing bounded rationality constitutes an additional consideration in classifying DUE models. Similarly elastic travel demand has long been embraced as a critical component of any general theory of static traffic assignment. The use of elastic travel demand models within a DUE modeling framework is discussed in Friesz et al. (1993) who describe in prose how to extend the results for fixed trip tables to a new formulation employing inverse demand functions. The technical details of that analysis were first carried out by Friesz and Meimand (2014) using differential variational inequality theory, which is an extension of optimal control theory, as presented in Friesz (2010). Subsequently (Han et al., 2015a) analyzed elastic demand dynamic user equilibrium (E-DUE) using functional analysis to address the important notions of existence and algorithm convergence. Other authors have also contributed to the study of DUE and DUE-like models and algorithms that accommodate elastic travel demand. Among these are Ran and Boyce (1996a), Cantarella (1997), Yang and Huang (1997), Yang and Meng (1998), Bliemer and Bovy (2003), Bellei et al. (2005), Bellei et al. (2006), and Szeto and Lo (2006).

1.8 Unresolved/Partially Resolved Fundamental Challenges In the review of analytical DUE modeling presented above, some critical modeling issues may be pointed out. The key issues are the following: 1. 2. 3. 4. 5.

effective path delay operators and convergence; the role of big data and inverse modeling; en route updating of destination, trip chaining, and trip purpose; feedback solutions; topological and mechanism network design

This book is meant to take steps toward meeting these challenges for so-called analytical dynamic user equilibrium models.

References and Suggested Reading

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Nahapetyan, A., & Lawphongpanich, S. (2007). Discrete-time dynamic traffic assignment models with periodic planning horizon: System optimum. Journal of Global Optimization, 38(1), 41– 60. Newell, G.F. (1993a). A simplified theory of kinematic waves in highway traffic, Part I: General theory. Transportation Research Part B, 27(4), 281–287. Newell, G. F. (1993b). A simplified theory of kinematic waves in highway traffic, Part II: Queuing at freeway bottlenecks. Transportation Research Part B, 27(4), 289–303. Newell, G. F. (1993c). A simplified theory of kinematic waves in highway traffic, Part III: Multidestination flows. Transportation Research Part B, 27(4), 305–313. Nie, X., & Zhang, H. M. (2005). A comparative study of some macroscopic link models used in dynamic traffic assignment. Networks and Spatial Economics, 5, 89–115. Nie, Y., & Zhang, H. M. (2008). A variational inequality formulation for inferring dynamic origindestination travel demands. Transportation Research Part B, 42 (7–8), 635–662. Nie, Y., & Zhang, H. M. (2010). Solving the dynamic user optimal assignment problem considering queue spillback. Networks and Spatial Economics, 10(2), 1–23. Nguyen, S. (1984). Estimating origindestination matrices from observed flows. In Florian, M. (Ed.), Transportation planning models (p. 363380). Amsterdam: Elsevier Science Publishers. Osorio, C., Flotterod, G., & Bierlaire, M. (2011). Dynamic network loading: A stochastic differentiable model that derives link state distributions. Transportation Research Part B, 45(9), 1410–1423. Pang, J. S., & Stewart, D. E. (2008). Differential variational inequalities. Mathematical Programming A, 113(2), 345–424. Pang, J. S., Han, L., Ramadurai, G., & Ukkusuri, S. (2012). A continuous-time linear complementarity system for dynamic user equilibria in single bottleneck traffic flows. Mathematical Programming, 133(1–2), 437–460. Peeta, S., & Ziliaskopoulos, A. K. (2001). Foundations of dynamic traffic assignment: The past, the present and the future. Networks and Spatial Economics, 1(3–4), 233–265. Perakis, G., & Roels, G. (2006). An analytical model for traffic delays and the dynamic user equilibrium problem. Operations Research, 54(6), 1151–1171. Rakha, H., & Tawfik, A. (2009) Traffic networks: Dynamic traffic routing, assignment, and assessment. Encyclopedia of complexity and systems science (pp. 9429–9470). Springer: New York Ramadurai, G., Ukkusuri, S. V., Zhao, J., & Pang, J. S. (2010). Linear complementarity formulation for single bottleneck model with heterogeneous commuters. Transportation Research Part B, 44(2), 193–214. Ran, B., & Boyce, D. E. (1996a). Modelling dynamic transportation networks: An intelligent transportation system oriented approach (p. 356), New York: Springer. Ran, B., & Boyce, D. E. (1996b). A link-based variational inequality formulation of ideal dynamic user-optimal route choice problem. Transportation Research Part C, 4(1), 1–12. Ran, B., Boyce, D. E., & LeBlanc, L. J. (1993). A new class of instantaneous dynamic user-optimal traffic assignment models. Operations Research, 41(1), 192–202. Ran, B., Hall, R. W., & Boyce, D. E. (1996b). A link-based variational inequality model for dynamic departure time/route choice. Transportation Research Part B, 30(1), 31–46. Richards, P. I. (1956). Shockwaves on the highway. Operations Research, 4, 42–51. Smith, M. J. (1993). A new dynamic traffic model and the existence and calculation of dynamic user equilibria on congested capacity-constrained road networks. Transportation Research Part B, 27(1), 49–63. Smith, M. J., & Wisten, M. B. (1995). A continuous day-to-day traffic assignment model and the existence of a continuous dynamic user equilibrium. Annals of Operations Research, 60(1), 59–79. Song, W., Han, K., Wang, Y., Friesz, T. L., & del Castillo, E. (2017). Statistical metamodeling of dynamic network loading. Transportation Research Part B. https://doi.org/10.1016/j.trb.2017. 08.018.

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Szeto, W. Y. (2013). Cell-based dynamic equilibrium models. In Advances in Dynamic Network Modeling in Complex Transportation Systems (pp. 163–192). Springer: New York. Szeto, W. Y., & Lo, H. K. (2004). A cell-based simultaneous route and departure time choice model with elastic demand. Transportation Research Part B, 38(7), 593–612. Szeto, W. Y., & Lo, H. K. (2005a) Dynamic traffic assignment: Review and future research directions. Journal of Transportation Systems Engineering and Information Technology, 5(5), 85–100. Szeto, W. Y., & Lo, H. K. (2005b) Properties of dynamic traffic assignment with physical queues. Journal of the Eastern Asia Society for Transportation Studies, 6, 2108–2123. Szeto, W. Y., & Lo, H. K. (2006). Dynamic traffic assignment: Properties and extensions. Transportmetrica, 2(1), 31–52. Szeto, W. Y., & Wong, S. C. (2012). Dynamic traffic assignment: Model classifications and recent advances in travel choice principles. Central European Journal of Engineering, 2(1), 1–18. Szeto, W. Y., Wang, Y., & Han, K. (2015). Bounded rationality in dynamic traffic assignment. In Rasouli, S., & Timmermans, H. (Eds.), Bounded rational choice behaviour: Applications in transport. Bingley: Emerald Group Publishing. Tong, C., & Wong, S. (2000). A predictive dynamic traffic assignment model in congested capacityconstrained road networks. Transportation Research Part B, 34(8), 625–644. Wardrop, J. G. (1952). Some theoretical aspects of road traffic research. ICE Proceedings: Engineering Divisions, 1(3), 325–362. Thomas Telford. Vickrey, W. S. (1963). Pricing in urban and suburban transport. The American Economic Review, 53(2), 452–465. Vickrey, W. S. (1969). Congestion theory and transport investment. The American Economic Review, 59(2), 251–261. Vickrey, W. S. (1968). Congestion charges and welfare. Journal of Transport Economics and Policy, 107–118. Viti, F., & Tampère, C. (2010). Dynamic traffic Assignment: Recent advances and new theories towards real time applications and realistic travel behaviour (Editorial). Cheltenham: Edward Elgar Publishing. Wie, B. W., Tobin, R. L., Friesz, T. L., & Bernstein, D. (1995). A discrete time, nested cost operator approach to the dynamic network user equilibrium problem. Transportation Science, 29(1), 79– 92. Wu, J. H., Chen, Y., & Florian, M. (1998). The continuous dynamic network loading problem: A mathematical formulation and solution method. Transportation Research Part B, 32(3), 173– 187. Wu, J., Sun, H., Wang, D. Z., Zhong, M., Han, L., & Gao, Z. (2013). Bounded-rationality based day-to-day evolution model for travel behavior analysis of urban railway network. Transportation Research Part C, 31, 73–82. Xu, Y. W., Wu, J. H., Florian, M., Marcotte, P., & Zhu, D. L. (1999). Advances in the continuous dynamic network loading problem. Transportation Science, 33, 341–353. Yang, H., & Hai-Jun, H. (1997). Analysis of the time-varying pricing of a bottleneck with elastic demand using optimal control theory. Transportation Research Part B, 31(6), 425–440 Yang, H., & Meng, Q. (1998). Departure time, route choice and congestion toll in a queuing network with elastic demand. Transportation Research Part B, 32(4), 247–260. Yao, T., Friesz, T. L., Wei, M., & Yin, M. (2010). Congestion derivatives for a traffic bottleneck. Transportation Research Part B, 44(10), 1149–1165. Yao, T., Wei, M., Zhang, B., Friesz, T. L. (2012). Congestion derivatives for a traffic bottleneck with heterogeneous commuters. Transportation Research Part B, 46(10), 1454–1473. Yperman, I., Logghe, S., & Immers, L. (2005). The link transmission model: An efficient implementation of the kinematic wave theory in traffic networks, Advanced OR and AI Methods in Transportation. In Proceedings of the 10th EWGT meeting and 16th Mini-EURO conference (pp. 122–127). Poznan, Poland: Publishing House of Poznan University of Technology. Zhu, D. L., & Marcotte, P. (2000). On the existence of solutions to the dynamic user equilibrium problem. Transportation Science, 34(4), 402–414. Ziliaskopoulos, A. K. (2000). A linear programming model for the single destination system optimal dynamic traffic assignment problem. Transportation Science, 34(1), 37–49.

Chapter 2

Mathematical Preliminaries

Central to the presentation and understanding of dynamic user equilibrium models is a sound mathematical foundation in a number of inter-related topics. This chapter aims to provide a quick overview of knowledge essential to the formulation, analysis, and computation of DUE models. These subjects include functional analysis, the calculus of variations, optimal control theory, mathematical programming, scalar conservation laws, and variational theory. Due to space limitations, the topics covered have been carefully selected and the style of their presentation is parsimonious. More comprehensive coverage may be found in the references at the end of this chapter.

2.1 Selected Topics in Functional Analysis 2.1.1 Hilbert Spaces We begin with inner product space on the field .R of real numbers, which is a vector space with an additional structure called an inner product. The inner product associated with each pair of vectors x and y, denoted .x, y, is a scalar function that satisfies • Symmetry: .

x, y = y, x

• Bi-linearity: .

ax, y = x, ay = a x, y x + y, z = x, y + y, z

© Springer Nature Switzerland AG 2022 T. L. Friesz, K. Han, Dynamic Network User Equilibrium, Complex Networks and Dynamic Systems 5, https://doi.org/10.1007/978-3-031-25564-9_2

33

34

2 Mathematical Preliminaries

• Positive-definiteness: .

x, x ≥ 0,

x, x = 0 ⇔ x = 0

An inner product space has a naturally defined norm based on the inner product of the space:  x, x . x = This is well-defined by the positive-definiteness of the inner product. The norm is thought of as the length of the vector x. Such a norm defines the distance between two points x and y in the vector space, namely, d(x, y) = x − y

.

With the distance function defined above, the inner product space becomes a metric space, that is, a space equipped with a measure of “distance” among its points. Definition 2.1 (Complete Metric Space) A metric space .M is called complete if every Cauchy sequence of points in .M has a limit that is also in .M. Definition 2.2 (Hilbert Space) A Hilbert space .H is a real inner product space that is also a complete metric space with respect to the distance induced by the inner product. Notice that a Hilbert space can be finite-dimensional (such as the Euclidean space) or infinite-dimensional (such as function spaces). A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. If we let .H be a Hilbert space and let .x, y ∈ H be any two non-zero vectors, then the angle between them is defined to be θ (x, y) =

.

x, y x · y

Let S be a subset of a Hilbert space .H, then the set of vectors orthogonal to S is defined via . S ⊥ = {x ∈ H : x, s = 0 ∀s ∈ S}

.

It is easily verifiable that .S ⊥ is a closed subspace of .H and itself forms a Hilbert space. Now assume V is a closed subspace of .H, then .V ⊥ is called the orthogonal complement of V . It is a well-known fact that each .x ∈ H can be uniquely decomposed into .x = v + u where .v ∈ V and .u ∈ V ⊥ . In view of this result, we call the linear operator .PV (·) : H → V , x → v the orthogonal projection onto V.

2.1 Selected Topics in Functional Analysis

35

The orthogonal projection .PV (·) is a minimum norm projection in the following sense: x − PV (x) ≤ y − PV (x)

.

∀y ∈ V

The notion of minimum norm projection does not require V to be a subspace. Instead, given any non-empty and closed set .S ⊂ H, the minimum norm projection operator, denoted by .PS (·), is defined to be PS (x) = argmin {s − x : s ∈ S}

.

∀x ∈ H

2.1.2 Topological Vector Spaces A topological vector space is one of the basic structures investigated in functional analysis. Such a space blends a topological structure with the algebraic concept of a vector space. The following is a precise definition: Definition 2.3 (Topological Vector Space) A topological vector space X is a vector space over a topological field .F (usually the real or complex numbers with their standard topologies) endowed with a topology such that the vector addition map .X × X → X and the scalar multiplication map .F × X → X are continuous functions. As a consequence of Definition 2.3, all normed vector spaces, and therefore all Banach spaces and Hilbert spaces, are examples of topological vector spaces. Also important is the notion of a seminorm: Definition 2.4 (Seminorm) A seminorm on a vector space X is a real-valued function p on X such that (a) .p(x + y) ≤ p(x) + p(y) (b) .p(α x) = |α| p(x) for all x and y in X and all scalars .α. Definition 2.5 (Local Convexity) A locally convex space is defined to be a vector space X along with a family of seminorms .{pi }i∈I on X. As part of our review, we make note of the following essential knowledge: Example 2.1 The space of square-integrable real-valued functions on a compact interval .[a, b], denoted by .L2 [a, b], is a locally convex topological vector space. Example 2.2  2 m The m-fold product of the spaces of square-integrable functions L [a, b] is a locally convex topological vector space.

.

Definition 2.6 (Dual Space) The dual space .X∗ of a vector space X is the space of all continuous linear functions on X.

36

2 Mathematical Preliminaries

Another key property we consider without proof is: Example 2.3 The dual space of .Lp [a, b] for .1 < p < ∞ has a natural isomorphism with .Lq [a, b] where q is such that .1/p + 1/q = 1. In particular, the dual space of 2 2 .L [a, b] is again .L [a, b].

2.1.3 Compactness Definition 2.7 (Compactness) A topological space X is called compact if each of its open covers has a finite cover. This means that for an arbitrary collection of open sets .

{Ua }a∈A

such that X ⊂



.

Ua

a∈A

there is a finite subset I ⊂ A such that X ⊂



.

Ui

i∈I

Definition 2.8 (Compactness of Subsets) A subset K of a topological space X is called compact if it is compact in the induced topology. This means that for an arbitrary collection of open sets .

{Ua }a∈A

such that K ⊂



.

Ua

a∈A

there is a finite subset I ⊂ A such that K ⊂



.

Ui

i∈I

In general topology, there is another important notion that is closely related to compactness: sequential compactness. In prose, a space is sequentially compact if every infinite sequence has a convergent subsequence. For general topological

2.1 Selected Topics in Functional Analysis

37

spaces, the notions of compactness and sequential compactness are not equivalent. However, they are equivalent for metric spaces. Proposition 2.1 In a metric space, the notions of compactness and sequential compactness are equivalent. Example 2.4 (Bolzano-Weierstrass Theorem) Each bounded sequence in the Euclidean space Rn has a convergent subsequence. In other words, a subset of Rn is sequentially compact (thus compact) if and only if it is bounded and closed. Example 2.5 (A Non-compact Subset) Consider the Hilbert space L2 [0, 1] consisting of square-integrable functions on the real interval [0, 1]. The inner product of two functions, f (·) and g(·) ∈ L2 [0, 1], is  .

1

f, g =

f (x) · g(x) dx

(2.1)

0

Given any constant M > 0, let the subset SM of L2 [0, 1] be defined as follows. . SM =

.

  f ≥0:

1

 f (x) dx = M

⊂ L2 [0, 1]

(2.2)

0

We claim that SM defined as such is not compact in L2 [0, 1] with respect to the metric induced by the inner product (2.1). Indeed, let us define the sequence of functions fn ∈ SM for n ≥ 0 as follows. fn (x) =

.

M · 2n

x ∈ [0, 1/2n ]

0

x ∈ (1/2n , 1]

For any m > n ≥ 0, a simple calculation shows that  .

fm − fn 2 = fm − fn , fm − fn  =

1

(fm (x) − fn (x))2 dx ≥ M 2 ·2n ≥ qM 2

0

This means that within the sequence {fn } , n ≥ 0, the distance between any two elements is greater than or equal to M. Thus there cannot be any convergent subsequence. We conclude that SM is not sequentially compact and thus not compact. Later in Chap. 5, where we provide existence results for the simultaneous route-and-departure-time choice dynamic user equilibrium in continuous time, the fact that SM defined in (2.2) is not compact poses a major challenge in the existence proof. Such a difficulty stems from the fact that known results suitable for proving existence of dynamic user equilibrium, such as Brouwer’s fixed point theorem and its equivalent forms, cannot be directly applied without the assurance of compactness of the underlying feasible set of path flows. A more detailed

38

2 Mathematical Preliminaries

explanation of the mathematical apparatus used to overcome such a difficulty will be presented in Chap. 5.

2.1.4 The Contraction Mapping Theorem The contraction mapping theorem is one of the most fundamental and useful results in functional analysis. It has been used extensively to construct solutions of linear and nonlinear equations. A contraction mapping on a metric space .M with distance function .d(·, ·) is a function f from .M to itself, with the property that there exists some .α < 1 such that, for all .x, y ∈ M, we have   d f (x), f (y) ≤ α d(x, y)

.

The contraction mapping theorem is that every contraction mapping has a fixed point. One form of the contraction mapping theorem is the following: Theorem 2.1 (Contraction Mapping Theorem) Let .M be a metric space. If f : M → M is a contraction mapping, then there is exactly one point .x ∈ M such that

.

f (x) = x

.

A somewhat more general version of the contraction mapping theorem is provided by Bressan and Piccoli (2007) and is presented below. Theorem 2.2 (Generalized Contraction Mapping Theorem) Let X be a Banach space, .Θ a metric space, and let .Φ : Θ × X → X be a continuous mapping such that, for some .κ < 1, we have .

Φ(θ, x) − Φ(θ, y) ≤ κ x − y

∀ θ, x, y

(2.3)

Then, for each .θ ∈ Θ, there exists a unique fixed point .x(θ ) ∈ X such that   x(θ ) = Φ θ, x(θ )

.

(2.4)

The map .θ → x(θ ) is continuous. Moreover, for any .θ ∈ Θ, y ∈ X, one has .

y − x(θ ) ≤

1 y − Φ(θ, y) 1−κ

(2.5)

2.2 Nonlinear Programming

39

2.2 Nonlinear Programming The primary intent of this section is to introduce the reader to the theoretical foundations of nonlinear programming. Particularly important are the notions of local and global optimality in mathematical programming, the Kuhn-Tucker necessary conditions for optimality in nonlinear programming, and the role played by convexity in making necessary conditions sufficient.

2.2.1 Nonlinear Program Defined We are presently interested in a type of optimization problem known as a finitedimensional mathematical program, namely: find a vector .x ∈ Rn that satisfies

.

min f (x) s.t. h(x) = 0 g(x) ≤ 0

⎫ ⎬ ⎭

(2.6)

where

.

x =

(x1 , . . . , xn )T ∈ Rn

f (·) :

Rn → R1

g(x) = (g1 (x), . . . , gm (x))T : Rn → Rm h(x) = (h1 (x), . . . , hq (x))T : Rn → Rq

We call the .xi for. i ∈ {1, 2, . . . , n} decision variables, .f (x) the objective function, h(x) = 0 the equality constraints, and .g(x) ≤ 0 the inequality constraints. Because the objective and constraint functions will in general be nonlinear, we shall consider (2.6) to be our canonical form of a nonlinear mathematical program (NLP). The feasible region for (2.6) is

.

X ≡ {x : g(x) ≤ 0, h(x) = 0} ⊂ Rn ,

.

(2.7)

which allows us to state (2.6) in the form .

min f (x) s.t. x ∈ X



The pertinent definitions of optimality for NLP are:

(2.8)

40

2 Mathematical Preliminaries

Definition 2.9 (Global Minimum) Suppose .x ∗ ∈ X and .f (x ∗ ) ≤ f (x) for all ∗ ∗ .x ∈ X. Then .f (x) achieves a global minimum on X at .x , and we say .x is a global minimizer of .f (x) on X. ∗ ∈ X and there exists an . > 0 Definition 2.10 (Local Minimum) Suppose .x

 ∗ ∗ such that .f (x ) ≤ f (x) for all .x ∈ N (x ) ∩ X , where .N (x ∗ ) is a ball of radius . > 0 centered at .x ∗ . Then .f (x) achieves a local minimum on X at .x ∗ , and we say .x ∗ is a local minimizer of .f (x).

In practice, we will often relax the formal terminology of Definitions 2.9 and 2.10 and refer to .x ∗ as a global minimum or a local minimum, respectively.

2.2.2 The Fritz John Conditions The fundamental theorem on necessary conditions is: Theorem 2.3 (Fritz John Conditions) Let .x ∗ be a (global or local ) minimum of .

min f (x) s.t. x ∈ F = {x ∈ X0 : g(x) ≤ 0, h(x) = 0} ⊂ Rn

where .X0 is a nonempty open set, .g : Rn −→ Rm , and .h : Rn −→ Rq . Assume that .f (x), .gi (x) for .i = 1, . . . , m and .hi (x) for .i = 1, . . . , q have continuous first derivatives everywhere on X. Then there must exist multipliers .μ0 ∈ R1+ , .μ = ∗ ∗ ∗ T q (μ1 , . . . , μm )T ∈ Rm + , and .λ = (λi , . . . ; λq ) ∈ R such that .

μ0 ∇f (x ∗ ) +

m

i=1 μi ∇gi (x

μi gi (x ∗ ) = 0 μi ≥ 0

∗) +

q

i=1 λi ∇hi (x

∗)

∀i ∈ {1, . . . , m} ∀ i ∈ [0, m]

(μ0 , μ, λ) = 0 ∈ Rm+q+1 Proof A formal proof of their validity is given in Friesz (2010).

=0.

(2.9)

.

(2.10)

.

(2.11) (2.12) 

Conditions (2.9), (2.10), (2.11), and (2.12) together with .h(x) = 0 and .g(x) ≤ 0 are called the Fritz John conditions.

2.2.3 The Kuhn-Tucker Conditions Under the linear independence constraint qualification and some other basic assumptions, the Kuhn-Tucker identity and the complementary slackness conditions

2.2 Nonlinear Programming

41

form, together with the original mathematical program’s constraints, a valid set of necessary conditions. The following theorem is a precise description: Theorem 2.4 (Kuhn-Tucker Conditions) Let .x ∗ ∈ F be a local minimum of .

min f (x) s.t. x ∈ F = {x ∈ X0 : g(x) ≤ 0, h(x) = 0}

where .X0 is a nonempty open set in .Rn . Assume that .f (x), .gi (x) for .1 ≤ i ≤ m and .hi (x) for .1 ≤ i ≤ q have continuous first derivatives everywhere on .F and that the gradients of binding constraint functions are linearly independent. Then there must exist multipliers . μ = (μ1 , . . . , μm )T ∈ Rm and .λ = (λ1 , . . . , λq )T ∈ Rq such that ∇f (x ∗ ) +

m 

.

μi ∇gi (x ∗ ) +

i=1

q 

λi ∇hi (x ∗ ) = 0

(2.13)

1≤i≤m

(2.14)

i=1

μi gi (x ∗ ) = 0

.

μi ≥ 0

.

1≤i≤m

Proof We refer the reader to Friesz (2010) for a formal proof.

(2.15) 

The Kuhn-Tuck conditions will later be applied in Sect. 3.5 to derive optimality conditions for finite-dimensional variational inequalities, which are common mathematical formulations of dynamic user equilibria.

2.2.4 Kuhn-Tucker Conditions Sufficient The concept of convexity plays an important role in the understanding of the notion of sufficiency. First, consider the following four definitions: Definition 2.11 (Convex Set) A set .X ⊆ Rn is convex if for any two vectors 1 2 .x , x ∈ X and any scalar .λ ∈ [0, 1] the vector x = λx 1 + (1 − λ)x 2

.

(2.16)

also lies in X. Definition 2.12 (Strictly Convex Set) A set .X ⊆ Rn is strictly convex if for any two vectors .x 1 and .x 2 in X and any scalar .λ ∈ (0, 1) the point x = λx 1 + (1 − λ)x 2

.

(2.17)

42

2 Mathematical Preliminaries

lies in the interior of X. Definition 2.13 (Convex Function) A scalar function .F (x) is a convex function defined over a convex set .X ⊆ Rn if for any two vectors .x 1 , x 2 ∈ X F (λx 1 + (1 − λ)x 2 ) ≤ λF (x 1 ) + (1 − λ)F (x 2 )

.

∀ λ ∈ [0, 1]

(2.18)

Definition 2.14 (Strictly Convex Function) In the above, .F (x) is strictly convex if the inequality is a strict inequality .( t0 while t0 , tf ⊂ R1+ . If a particular curve .x(t) satisfies the initial conditions (2.34) as well as the terminal conditions (2.35), we say it is an admissible trajectory. A trajectory that maximizes or minimizes the criterion in (2.33) is called an extremal of .J (x). An admissible trajectory that minimizes .J (x) is a solution of the variational problem (2.33), (2.34), and (2.35). Also, the reader should note that in (.2.33) the objective .J (x) should be referred to as the criterion functional, never as the “criterion function.” This is because .J (x) is actually an operator, and we are seeking as a solution the function .x (t); this distinction is sometimes explained by saying a “functional is a function of a function.” The variation of the decision function .x (t), written as .δx (t), obeys

.

dx (t) = δx (t) + x˙ (t) dt

.

(2.38)

In other words, the total differential of .x (t) is its variation .δx (t) plus the change in the variable attributed solely to time, namely, .x˙ (t) dt. To understand the variation of the criterion functional n we denote the change in the functional arising from  .J (x), the increment .h ∈ C 1 t0 , tf by

J (h) ≡ J (x + h) − J (x)

.

n  for each .x ∈ C 1 t0 , tf . This allows us to make the following definition:

(2.39)

2.3 Calculus of Variations

47

Definition 2.15 (Differentiability and Variation of a Functional) If

J (h) = δJ (h) + ε h

(2.40)

.

n  where, for any given .x ∈ C 1 t0 , tf , .δJ (h) is a linear functional of .h ∈  1 n C t0 , tf and .ε −→ 0 as .h −→ 0, .J (x) is said to be differentiable and .δJ (h) is called its variation (for the increment h). This definition is conveniently summarized by saying that the variation of the functional .J (x) is the principal linear part of the change . J (h). Note that the variation is dependent on the increment taken for each x. Furthermore, since the variation is the principal linear part, it may be found by retaining the linear terms of a Taylor series expansion of the criterion functional about the point x. To illustrate let us consider the functional  tf .J (w1 , w2 ) = F (w1 , w2 ) dt. (2.41) t0

w1 (t0 ), w2 (t0 ) fixed.

(2.42)

w1 (tf ), w2 (tf ) fixed

(2.43)

where for convenience we take .w1 , .w2 , and .F (·, ·) to be scalars. The change in this functional for the increment .h = (h1 , h2 )T is

J (h1 , h2 ) = J (w1 + h1 , w2 + h2 ) − J (w1 , w2 ).

(2.44)

.

 =

tf t0

 ∂F (w1 , w2 ) F (w1 , w2 ) + [(w1 + h1 ) − w1 ] . ∂w1

 ∂F (w1 , w2 ) [(w2 + h2 ) − w2 ] − F (w1 , w2 ) dt + ε h ∂w2

+ .

 =

(2.45)

tf



t0



∂F (w1 , w2 ) ∂F (w1 , w2 ) h1 + h2 dt + ε h ∂w1 ∂w2

(2.46) (2.47)

It is immediate that  δJ (h1 , h2 ) =

tf

.

t0



 ∂F (w1 , w2 ) ∂F (w1 , w2 ) h1 + h2 dt ∂w1 ∂w2

(2.48)

48

2 Mathematical Preliminaries

If we identify the decision variable variations .δw1 and .δw2 with the increments .h1 and .h2 , respectively, this last expression becomes  δJ (h1 , h2 ) =

tf

.

t0



 ∂F (w1 , w2 ) ∂F (w1 , w2 ) δw1 + δw2 dt, ∂w1 ∂w2

(2.49)

which is a chain rule for the calculus of variations. Expression (2.49) is a specific instance of the following variational calculus general chain rule: the variation of the functional (2.33) obeys δJ (x) =

n  

tf

.

i=1

t0



 ∂f0 ∂f0 δxi + δ x˙i dt ∂xi ∂ x˙i

(2.50)



where .x (t) ∈ Rn for each instant of time .t ∈ t0 , tf . We reiterate that, in the language we have introduced, .δJ (x) is the variation of the functional .J (x).

2.4 Optimal Control We limit our discussion to continuous optimal control due to the prevalence and importance that continuous time plays in dynamic traffic assignment. See Friesz (2010) for an overview of discrete optimal control. In the theory of optimal control, we are concerned with extremizing (maximizing or minimizing) a criterion functional subject to constraints. Both the criterion and the constraints are articulated in terms of two types of variables: control variables and state variables. The state variables obey a system of first-order ordinary differential equations whose right-hand sides typically depend on the control variables; initial values of the state variables are either specified or meant to be determined in the process of solving a given optimal control problem. Consequently, when the control variables and the state initial conditions are known, the state dynamics may be integrated and the state trajectories found. In this sense, the state variables are not really the decision variables; rather, the control variables are the fundamental decision variables. For reasons that will become clear, we do not require the control variables to be continuous; instead we allow the control variables to exhibit jump discontinuities. Furthermore, the constraints of an optimal control problem may include, in addition to the state equations and state initial conditions already mentioned, constraints expressed purely in terms of the controls, constraints expressed purely in terms of the state variables, and constraints that involve both control variables and state variables. The set of piecewise continuous controls satisfying the constraints imposed on the controls is called the set of admissible controls. Thus, the admissible controls are roughly analogous to the feasible solutions of a mathematical program.

2.4 Optimal Control

49

Consider now the following canonical form of the continuous-time optimal control problem with pure control constraints: 

  criterion : min J [x (t) , u (t)] = K x tf , tf +



tf

f0 [x (t) , u (t) , t] dt

.

t0

(2.51)

subject to the following: state dynamics :

.

dx = f (x (t) , u (t) , t) . dt

initial conditions : x (t0 ) = x0 ∈ Rm t0 ∈ R1. 

  tf ∈ R1. terminal conditions :  x tf , tf = 0 

control constraints : u (t) ∈ U ∀t ∈ t0 , tf

(2.52) (2.53) (2.54) (2.55)



where for each instant of time .t ∈ t0 , tf ⊂ R1+ : x (t) = (x1 (t) , x2 (t) , . . . , xn (t))T .

(2.56)

u (t) = (u1 (t) , u2 (t) , . . . , um (t))T .

(2.57)

.

f0 : Rn × Rm × R1 −→ R1.

(2.58)

f : Rn × Rm × R1 −→ Rn.

(2.59)

K : Rn × R1 −→ R1.

(2.60)

 : Rn × R1 −→ Rr

(2.61)

We will use the notation .OCP(f0 , f, K, , U, x0 , t0 , tf ) to refer to the above canonical optimal control problem. We assume the functions .f0 (., ., ), . (., .), .K(., .), and .f (., ., .) are everywhere once continuously differentiable with respect to their arguments. In fact, we employ the following definition: Definition 2.16 (Regularity for .OCP(f0 , f, K, , U, x0 , t0 , tf )) We shall say optimal control problem .OCP(f0 , f, K, , U, x0 , t0 , tf ) defined by (2.51), (2.52), (2.53), (2.54), and (2.55) is regular provided .f (x, u, .), .f0 (x, u, .), .[x(tf ), tf ], and .K[x(tf ), tf ] are everywhere once continuously differentiable with respect to their arguments. We need to formally define the notion of an admissible solution for OCP(f0 , f, K, , U, x0 , t0 , tf )

.

That definition is the following:

50

2 Mathematical Preliminaries

Definition 2.17 (Admissible Control Trajectory) We say that the control trajectory .u(t) is admisible relative

to .OCP(f0 , f, K, , U, x0 , t0 , tf ) if it is piecewise continuous for all time .t ∈ t0 , tf and .u ∈ U . Note that the initial time and the terminal time may be unknowns in the continuoustime  optimal control problem. Moreover, the initial values .x (t0 ) and final values .x tf may be unknowns. Of course, the initial and/or final values may also be stipulated. The unkowns are the state variables x and the control variables u. It is critically important to realize that the state variables will generally be completely determined when the controls and initial states are known. Consequently, the “true” unknowns are the control variables u. Note also that we have not been specific about the vector spaces to which  

x = x (t) : t ∈ t0 , tf 

 u = u (t) : t ∈ t0 , tf

.

belong. This is by design, as we shall initially discuss the continuous-time optimal control problem by developing intuitive dynamic extensions of the notion of stationarity and an associated calculus for variations of .x (t) and .u (t).

2.4.1 The State Operator We begin by considering the control vector  m u ∈ L2 [t0 , tf ]

.

and associated operator    dy = f y(t), u(t), t , y(t0 ) = x0 , Γ [y(tf ), tf ] = 0 dt n  0 (2.62) ∈ C [t0 , tf ] 

x(u, t) = arg

.

where x0 ∈ Rn.

(2.63)

f : Rn × Rm × R1 −→ Rn.

(2.64)

Γ : Rn × R1 −→ Rn

(2.65)

.

2.4 Optimal Control

51

The entity .x(u, t) is to be interpreted as an operator, called the state operator, that tells us the state vector x for each control vector u and each time .t ∈ [t0 , tf ] ⊂ R when there are end point conditions that the state variables must satisfy. Working with this operator is, in effect, a supposition that a two-point boundary value problem involving the state variables has a solution for each control vector considered.1 Furthermore, we assume that every control vector is constrained to lie in a set  m U ⊂ L2 [t0 , tf ]

.

m  where . L2 [t0 , tf ] is the m-fold product of the space of square-integrable functions .L2 [t0 , tf ] with inner product defined by  .

 u, v =



tf

[u(t)]T v(t) dt.

(2.66)

t0

The superscript T stands for transpose of vectors. U is defined so as to ensure the terminal conditions imposed on the state variables may be reached from the initial conditions intrinsic to (2.62). To analyze the existence, continuity, and differentiability of the state operator, we rely on the following assumption.  .  (Assumption 1) The set .U = u(t) : t ∈ [t0 , tf ], u ∈ U ⊂ Rm of control values is compact. The function .f = f (x, u, t) is defined and continuous on the space n .R × U × R, continuously differentiable with respect to x, and satisfies   f (x, u, t) ≤ C,

.

  Dx f (x, u, t) ≤ L

(2.67)

for some constants .C, L, and all .x, u, t. The existence of the state operator is an essential issue we must explore. Let us begin with the following theorem to establish the existence of a solution to an ODE. Theorem 2.8 (Existence of the State Operator) Let us consider an initial value problem .

dx = f (x, u, t). dt

x(t0 ) = x0

1 Note that constraints on u are enforced separately. This definition of .x(u,

(2.68) (2.69)

t) is precisely that given by Minoux (1986) when analyzing optimal control problems from the point of view of infinite dimensional mathematical programming. Moreover, .x(u, t) should be thought of as a parametric representation of the state vector in terms of the controls. Note also that we do not actually have to explicitly solve for .x(u, t), as is made clear in our subsequent analysis; so in working with .x(u, t), we are not presuming existence of a solution of the variational inequality to be articulated below.

52

2 Mathematical Preliminaries

for .t ∈ [t0 , tf ]. Suppose .f (x, u, t) is Lipschitz continuous in x for all .t ∈ [t0 , tf ]. That is, the condition     f (x, u, t) − f (x, ˆ u, t) ≤ Lx − xˆ 

.

(2.70)

holds for all .x, x, ˆ and a constant .L ≥ 0. Let Assumption 1 hold, then the initial value problem (2.68)–(2.69) has a unique solution .x(t), t ∈ [t0 , tf ], for every given .u ∈ U . 

Proof See Walter (1988).

Theorem 2.8 states that given any .u ∈ U , a unique trajectory .x(u, ·) exists for all t ∈ [t0 , tf ]. As such, the state operator exists and is well-defined.

.

Theorem 2.9 (Continuity of the State Operator) Assumption 1 hold. Then   Let m n the map .u → x(u, ·) is continuous from . L2 [t0 , tf ] into . C 0 [t0 , tf ] . 

Proof A proof can be found in Bressan and Piccoli (2007).

Let us give a differentiability property of the state operator with respect to the control in the Gateaux sense. Theorem 2.10 (Differentiability of the State Operator w.r.t. the Control) In addition to Assumption 1, assume that f is continuously differentiable in an open neighborhood .V of .U. Let .u(·) ∈ U with corresponding trajectory .x(u, ·) be defined on .[t0 , tf ]. Then for every bounded measurable .Δu(·) and every .t ∈ [t0 , tf ], .x(u, ·) is G-differentiable with respect to u. That is, the derivative x(u + εΔu, t) − x(u, t) ε→0 ε

δx(u, Δu) ≡ lim

.

(2.71)

exists for every such .Δu. In particular,  δx(u, Δu) =

.

  M(t, x)Du f x(u, x), u(s), s · Δu(s) ds

where .Du f denotes the matrix of partial derivatives fundamental solution for the linearized problem

.

∂fi ∂uj

, and M is the matrix

  v(t) ˙ = Dx f x(u, t), u(t), t · v(t)

.

Proof The proof follows Bressan and Piccoli (2007).

(2.72)

(2.73) 

2.4 Optimal Control

53

2.4.2 Necessary Conditions for Continuous-Time Optimal Control Relying as it does on the variational notation introduced early in Sect. 2.3.2, our derivation of the optimal control necessary conditions in this section will be informal. To derive necessary conditions in such a manner, we will need the variation of the state vector x, denoted by .δx. We will make use of the relationship dx = δx + xdt ˙

.

(2.74)

that identifies .δx, the variation of x, as that part of the total change dx not attributable to time. Variations of other entities, such as u, are denoted in a completely analogous fashion. We start our derivation of optimal control necessary conditions by pricing out all constraints to obtain the Lagrangean  

 

  L = K x tf , tf + ν T  x tf , tf  tf   + f0 (x, u, t) + λT [f (x, u, t) − x] ˙ dt

.

(2.75)

t0

Using the variational calculus chain rule developed earlier, we may state the variation of the Lagrangean .L as       

  δL = t tf dtf + x tf dx tf + f0 tf dtf − f0 (t0 ) dt0  tf   Hx δx + Hu δu − λT δ x˙ dt +

.

(2.76)

t0

where H (x, u, λ, t) ≡ f0 (x, u, t) + λT f (x, u, t)

.

(2.77)

is the Hamiltonian and

     

   tf ≡ K x t f , tf + ν T  x tf , tf .

.

(2.78)

f0 (t0 ) ≡ f0 [x (t0 ) , u (t0 ) , t0 ] .  

     f0 tf ≡ f0 x tf , u tf , tf .

(2.79) (2.80)

t ≡

∂ ∂t

x ≡

∂ . ∂x

(2.81)

f0x ≡

∂f0 ∂x

fx ≡

∂f . ∂x

(2.82)

54

2 Mathematical Preliminaries

Hx ≡

∂H = (∇x H )T ∂x

Hu ≡

We next turn our attention to the term  tf    −λT δ x˙ dt = − .I ≡ t0

∂H = (∇u H )T ∂u

tf

λT

t0

 =−

tf

(2.83)

d (δx) dt dt

λT d (δx)

t0

appearing in (2.76). In particular, using the rule for integrating by parts2 this integral becomes  tf       T T dλT δx tf δx tf + .I = λ (t0 ) δx (t0 ) − λ t0

    = λ (t0 ) δx (t0 ) − λ tf δx tf + T

T



! dλT δx dt dt

tf t0

(2.84)

We also note that       δx tf = dx tf − x˙ tf dtf .

(2.85)

δx (t0 ) = dx (t0 ) − x˙ (t0 ) dt0

(2.86)

.

from the definition of a variation of the state vector. Using (2.84) in (2.76) gives           δL = t tf dtf + Tx tf dx tf + f0 tf dtf − f0 (t0 ) dt0  tf + [Hx δx + Hu δu] dt

.

t0

    + λ (t0 ) δx (t0 ) − λ tf δx tf + T

T



tf t0

2 Integration

" " by parts: . udv = uv − vdu.

! dλT δx dt dt

(2.87)

2.4 Optimal Control

55

Using (2.85) and (2.86) in (2.87) gives       

  δL = t tf dtf + x tf dx tf + f0 tf dtf − f0 (t0 ) dt0        + λT (t0 ) [dx (t0 ) − x˙ (t0 ) dt0 ] − λT tf dx tf − x˙ tf dtf

.

 +

tf



t0

dλT Hx + dt

!

 δx + Hu δu dt

(2.88)

It follows from (2.88), upon rearranging and collecting terms, that          δL = t tf + f0 tf + λT tf x˙ tf dtf

.

       + Tx tf − λT tf dx tf + λT (t0 ) dx (t0 )   − f0 (t0 ) + λT (t0 ) x˙ (t0 ) dt0  +

tf

   Hx + λ˙ T δx + Hu δu dt

(2.89)

t0

We see from (2.89) that, in order for .δL to vanish for arbitrary admissible variations, the coefficient of each individual differential and variation must be zero. That is, for the case of no explicit control constraints, .δL = 0 is ensured by the following necessary conditions for optimality: 1. state dynamics: dx = f (x (t) , u (t) , t) dt

(2.90)

H (t0 ) = 0 and λ (t0 ) = 0 ⇒ f0 [x (t0 ) , u (t0 ) , t0 ] = 0 .

(2.91)

.

2. initial time conditions: .

x (t0 ) = x0 ∈ Rm

(2.92)

3. adjoint equations: λ˙ = −Hx = −f0x − λT fx

.

(2.93)

4. transversality conditions:    

  

   λ tf = x tf = Kx x tf , tf + ν T x x tf , tf

.

(2.94)

56

2 Mathematical Preliminaries

5. terminal time conditions:

   t x tf , tf = 0.     −H tf = t tf

.

(2.95) (2.96)

where

              H tf ≡ f0 x tf , u tf , tf + λT tf f x tf , u tf , tf  

  

   t tf ≡ Kt x tf , tf +ν T t x tf , tf .

6. minimum principle.: Hu (x, u, λ, t) = 0

.

(2.97)

Note carefully that a two-point boundary value problem is an explicit part of these necessary conditions. That is to say, we need to solve a system of ordinary differential equations, namely, the original state dynamics (2.90) together with the adjoint equations (2.93), given the initial values of the state variables (2.92) and the transversality conditions (2.94) imposed on the adjoint variables at the terminal time; this will typically be the case even when the initial time.t0 , the terminal time .tf , and the initial state .x (t0 ) are fixed and the terminal state .x tf is free. Note also that when the initial time .t0 is fixed, we do not enforce (2.91), since .dt0 will vanish. To develop necessary conditions for the case of explicit control constraints, we invoke an intuitive argument. In particular, we argue that the total variation expressed by .δL must be nonnegative if the current solution is optimal; otherwise, there would exist a potential to decrease .L (and hence J ), and such a potential would not be consistent with having achieved a minimum. Since it is only the variation .δu that is impacted by the constraints .u ∈ U and which can no longer be arbitrary, we may invoke all the conditions developed above except the one requiring the coefficient of .δu to vanish; so, we require  δL =

tf

.

(Hu δu) dt ≥ 0

(2.98)

t0

In order for condition (2.98) to be satisfied for all admissible variations .δu, we require   Hu δu = Hu u − u∗ ≥ 0 ∀u ∈ U

.

where we have expressed the variation of u as δu = u − u∗ ,

.

(2.99)

2.4 Optimal Control

57

which describes feasible directions rooted at the optimal control solution .u∗ ∈ U when the set U is convex. Inequality (2.99) is the correct form of the minimum principle when there are explicit, pure control constraints forming a convex set U ; it is known as a variational inequality. The above discussion has been a constructive proof of the following result: Theorem 2.11 (Necessary Conditions for Continuous-Time Optimal Control ˙ and u are well defined and linear in their Problem) When the variations of x, .x, increments, the set of feasible controls U is convex, and regularity in the sense of Definition 2.16 obtains, the conditions (2.90), (2.91), (2.92), (2.93), (2.94), (2.95), (2.96), and (2.99) are necessary conditions for a solution of the optimal control problem .OCP(f0 , f, K, , U, x0 , t0 , tf ) defined by (2.51), (2.52), (2.53), (2.54), and (2.55).

2.4.3 Sufficiency in Optimal Control The necessary conditions considered in this chapter may only be used to find a globally optimal solution if we are able to uncover and compare all of the solutions of them. This is of course not in general possible. Consequently, we are interested, in this section, in regularity conditions that make the optimal control necessary conditions developed previously sufficient for optimality. There are two main types of sufficiency theorems employed in optimal control theory. We refer to these loosely as the Mangasarian and the Arrow theorems. Actually, Arrow’s original proof of his sufficiency theorem was incomplete although the theorem itself was correct. The correct proof of Arrow’s sufficiency theorem is generally attributed to Seierstad and Sydsæter (1977). Mangasarian’s theorem essentially states that, when no state-space constraints are present, the Pontryagin necessary conditions are also sufficient if the Hamiltonian is convex (when minimizing) with respect to both the state and the control variables. By contrast, the Arrow sufficiency theorem requires only that the Hamiltonian restated via the minimum principle (2.97) or (2.99) be convex with respect to the state variables.

2.4.3.1

The Mangasarian Theorem

We are interested in this section in proving one version of the Mangasarian (1966) sufficiency theorem for the continuous-time optimal control problem. This can be done with relative ease for the case of fixed initial and terminal times. We will additionally assume that there are no terminal time conditions and that the initial state is known and fixed. We will also assume that the Hamiltonian, when minimizing, is jointly convex in both the state variables and the control variables.

58

2 Mathematical Preliminaries

In particular, we study the following version of Mangasarian’s theorem articulated by Seierstad and Sydsæter (1977) and Seierstad and Sydsæter (1999): Theorem 2.12 (Restricted Mangasarian Sufficiency Theorem) Suppose the admissible pair .(x ∗ , u∗ ) satisfies all of the relevant continuous-time optimal control necessary conditions for .OCP(f0 , f, K, , U, x0 , t0 , tf ) when regularity in the sense of Definition 2.16 obtains, the set of feasible controls U is convex, the Hamiltonian H is jointly

convex   in x and u for all admissible solutions, .t0 and .tf are fixed, .x0 is fixed, .K x tf , tf = 0, and there are no terminal time conditions    = 0. Then any solution of the continuous-time optimal control . x tf , tf necessary conditions is a global minimum. Proof We follow the exposition of Seierstad and Sydsæter (1999) and begin the proof by noting that for .(x ∗ , u∗ ) to be optimal it must be that 

≡



tf

f0 (x, u, t) dt −

.

t0

tf

  f0 x ∗ , u∗ , t dt ≥ 0

∀ admissible (x, u)

t0

(2.100)

when minimizing. Moreover, the associated Hamiltonian is H = f0 + λT x˙

.

We note that (2.100) may be restated as 

=

tf

.

  H − H ∗ dt −

t0



tf

  λT x˙ − x˙ ∗ dt

(2.101)

t0

Since H is convex with respect to x and u, the tangent line underestimates, and we write H∗ +

.

 ∂H ∗   ∂H ∗  x − x∗ + u − u∗ ≤ H ∂x ∂u

or equivalently .

 ∂H ∗   ∂H ∗  x − x∗ + u − u∗ ≤ H − H ∗ ∂x ∂u

(2.102)

It follows from (2.101) and (2.102) that 

≥

tf

.

t0



  tf  ∂H ∗     ∂H ∗  ∗ ∗ x−x + u−u λT x˙ − x˙ ∗ dt dt − ∂x ∂u t0

(2.103)

2.4 Optimal Control

59

Using the adjoint equation .−dλT /dt = ∂H ∗ /∂x, this last result becomes 

   tf  ∂H ∗     dλT  ∗ ∗ − x−x + u−u λT x˙ − x˙ ∗ dt dt − dt ∂u t0

tf

≥

.

t0



tf

=−



t0



tf

=−

t0

  tf     dλT  ∂H ∗  ∗ T ∗ x − x + λ x˙ − x˙ u − u∗ dt dt + dt ∂u t0

 tf   d  T ∂H ∗  ∗ λ x−x u − u∗ dt dt + dt ∂u t0

   tf = − λT x − x ∗ + t0

tf

t0

 ∂H ∗  u − u∗ dt ∂u

(2.104)

Expression (2.104) allows us to write   

     

≥ − λT (t0 ) x (t0 ) − x ∗ (t0 ) + λT tf x tf − x ∗ tf

.

 +

tf t0

 ∂H ∗  u − u∗ dt ∂u

     = −λT (t0 ) [0] + 0 x tf − x ∗ tf +



tf t0

 =

tf

t0

 ∂H ∗  u − u∗ dt ≥ 0 ∂u

 ∂H ∗  u − u∗ dt ∂u (2.105)

where inequality (2.105) follows from the convexity of U and the fact the minimum  principle is satisfied. Thus . ≥ 0, and we have established optimality. Theorem 2.12 is easily extended to the case of mixed terminal conditions and a nontrivial salvage function. These generalizations are left as an exercise for the reader.

2.4.3.2

The Arrow Theorem

In Arrow and Kurz (1970), an alternative sufficiency theorem is presented, a theorem that is generally credited to Arrow. The Arrow theorem is based on a reduced form of the Hamiltonian obtained when the optimal control law derived from the minimum principle is employed to eliminate control variables from the Hamiltonian. Under appropriate conditions, if the reduced Hamiltonian is convex in the state variables, the necessary conditions are also sufficient. Seierstad and Sydsæter (1999) provide the following statement and proof of the Arrow result:

60

2 Mathematical Preliminaries

Theorem 2.13 (The Arrow Sufficiency Theorem) Let .(x ∗ , u∗ ) be an admissible pair for .OCP(f0 , f, K, , U, x0 , t0 , tf ) when regularity in the sense of Definition 2.16 obtains, the set of feasible controls is .U = R m , the Hamiltonian H is jointly  in x and u for all admissible solutions, .t0 and .tf are fixed,  .x0 is fixed,

 convex = 0, and there are no terminal time conditions . x tf , tf = 0. .K x tf , tf If there exists a continuous and piecewise continuously differentiable function T .λ = (λ1 , . . . , λn ) such that the following conditions are satisfied: λ˙ i =

.

−∂H ∗ , almost everywhere ∂xi

i = 1, . . . , n.

(2.106)



H (x ∗ , u, λ(t), t) ≥ H (x ∗ , u∗ , λ, t) for all u ∈ U and all t ∈ t0 , tf . (2.107) 

#(x, λ, t) = min H (x, u, λ, t) exists and is convex in x for all t ∈ t0 , tf H u∈U

(2.108) #(x, λ, t) is then .(x ∗ , u∗ ) solves .OCP(f0 , f, K, , U, x0 , t0 , tf ) for the given. If .H ∗ ∗ strictly convex in x for all t, then .x is unique (but .u is not necessarily unique). Proof Suppose .(x, u) and .(x ∗ , u∗ ) are admissible pairs and that .(x ∗ , u∗ ) satisfies the minimum principle and related necessary conditions. Optimality will be assured if we show that  tf  tf   . = f0 (x, u, t) dt − f0 x ∗ , u∗ , t dt ≥ 0 ∀ admissible (x, u) t0

t0

(2.109)

Suppose we are able to establish, for all admissible .(x, u), that H − H ∗ ≥ λ˙ T (x − x ∗ )

.

Then it is immediate that   tf   ∗ H − H dt − . = t0

tf

  λT x˙ − x˙ ∗ dt ≥ 0,

(2.110)

(2.111)

t0

which in turn implies (2.109). Consequently, it is enough to establish condi#, we have tion (2.110). From definition (2.108) for .H #∗ H∗ = H

.

# H ≥H Therefore #−H #∗ H − H∗ ≥ H

.

(2.112)

2.5 Differential Variational Inequalities

61

Consequently it suffices to prove #−H #∗ ≥ −λ˙ T (x − x ∗ ) H

(2.113)

.

#(x, λ, t) at .x ∗ . for any admissible x, an inequality that makes .−λ˙ a subgradient of .H To prove the existence of the subgradient, let us suppose #−H #∗ ≥ a T (x − x ∗ ), H

(2.114)

.

# is differentiable, it is immediate that again for all admissible x. If .H .

% $x % ∗ = ∇Hx∗ = −λ˙ ∇H x=x

(2.115)

From (2.112) and (2.114), we have H − H ∗ ≥ a T (x − x ∗ )

(2.116)

.

Consequently G(x) = H − H ∗ − a T (x − x ∗ ) ≥ 0

.

∀x

(2.117)

 for any .t ∈ t0 , tf and all admissible x. Note that since .G(x ∗ ) = 0, we know .x ∗ minimizes .G(x). Therefore .∇x G(x ∗ ) = 0; that is ∇x G(x ∗ ) = ∇H |x=x ∗ − a = 0,

.

(2.118)

Moreover, the adjoint equation compels .

− λ˙ = ∇H ∗ = a,

(2.119)

thereby establishing the existence of a subgradient, namely, .a = ∇H ∗ . It is then immediate from (2.116) and (2.119) that #−H #∗ ≥ a T (x − x ∗ ) = −λ(x ˙ − x ∗ ), H

.

which is recognized to be identical to (2.113) and completes the proof.

(2.120) 

2.5 Differential Variational Inequalities To articulate an adequately general differential variational inequality with controls, we must specify the function spaces associated with the key mappings that arise in such a problem formulation.

62

2 Mathematical Preliminaries

2.5.1 Problem Definition We begin by considering the control vector  m u ∈ L2 t0 , tf

.

and associated state operator  

  dy = f (y, u, t) , y (t0 ) = y0 , Ψ y tf , tf = 0 .x(u, t) = arg dt  n (2.121) ∈ H 1 t0 , tf 

where x0 ∈ Rn. (2.122)       n m n 



 f : H 1 t0 , tf × L2 t0 , tf × R1+ −→ L2 t0 , tf . (2.123)

.

Ψ : Rn × R1+ −→ Rr

(2.124)

 m and . L2 t0 , tf is the m-fold product of the space of square-integrable functions

 2 t ,t .L 0 f with inner product defined by  v, u =

tf

.

v T udt

(2.125)

t0

 n

 while . H1 t0 , tf is the .n-fold product of the Sobolev space .H1 t0 , tf ; the reader is referred to Evans (1995) for the definition of Sobolev spaces. The entity .x(u, t) is to be interpreted as an operator that tells

us the state vector x for each control vector u and each instant of time .t ∈ t0 , tf ⊂ R1+ when there are end point conditions that the state variables must satisfy. Working with this operator is, in effect, a supposition that the two-point boundary value problem involving the state variables has a unique solution for each control vector considered. That is, terminal states obeying the terminal constraints are reachable from the specified initial states for each admissible control. Note that constraints on u are enforced separately, so in working with .x(u, t) we are not presuming existence of a solution of the variational inequality to be articulated below. Moreover, unless other conditions are satisfied .x (u, t) is not a solution of the variational inequality considered in (2.126); rather it should be thought of as a parametric representation of the state vector in terms of the controls. Note also that we do not actually have to explicitly solve for .x(u, t), as is made clear in our subsequent analysis. The notion of a state operator .x (u, t) is precisely that used in Sect. 2.4.1 when analyzing optimal control problems and

2.5 Differential Variational Inequalities

63

is not original to us but has been employed by others; see, for example, Minoux (1986). Furthermore, we assume that every control vector is constrained to lie in a set  m U ⊂ L 2 t0 , tf ,

.

where U is defined to ensure the terminal conditions imposed on the state variables may be reached from the initial conditions intrinsic to (2.121). Given the operator (2.121), the variational inequality of interest to us takes the following form: .

find u∗ ∈ U such that F (x(u∗ , t), u∗ , t) , u − u∗  ≥ 0 ∀u ∈ U

 (2.126)

where   n  2 m m F : H 1 t0 , tf × L t0 , tf × R1+ −→ L2 t0 , tf

.

Note that, by virtue of the inner product (2.125), we may state the variational inequality (2.126) as  ∗  ∗ ∗ .F x(u , t), u , t , u − u  ≡



tf

 ∗ ∗ T   F x(u ), u , t u − u∗ ≥ 0

t0

We refer to (2.126) as a differential variational inequality (with explicit state equations and controls) and give it the symbolic name DVI(F, f, .Ψ , U, x.0 , t.0 , t.f ).

2.5.2 Regularity Conditions for DIV To analyze (2.126) we will rely on the following notion of regularity: Definition 2.18 (Regularity of DVI.(F, f, Ψ, U, x0 , t0 , tf )) We call DVI(F, f, .Ψ , U, x.0 , t.0 , t.f ) regular if: m  R1. .u ∈ U ⊆ L2 t0 , tf  n R2. .x ∈ . H1 t0 , tf m n  2  R3. .x (u, t) : L t0 , tf × R1+ −→ H1 t0 , tf exists and is unique, strongly continuous and G-differentiable for all admissible u; R4. .Ψ : Rn × R1+ −→ Rr is continuously differentiable with respect to x and t; n  2 m

  R5. .F : H1 t0 , tf × L t0 , tf × R1+ −→ L2 t0 , tf is continuous with respect m n  to x andu;   n R6. .f : H1 t0 , tf × L2 t0 , tf × R1+ −→ L2 t0 , tf is continuously differentiable with respect to x and u;

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2 Mathematical Preliminaries

R7. .x0 ∈ Rn , .t0 ∈ R1+ , and .tf ∈ R1++ are known and fixed; m  R8. .U ⊂ L2 t0 , tf is convex; and R9. there is a constant dual vector .υ ∈ Rr for the terminal constraints

   .Ψ x tf , tf = 0.

2.5.3 Necessary Conditions To develop necessary conditions for solutions of DVI(F, f, .Ψ , U, x.0 , t.0 , t.f ), we note that (2.126) may be restated as the following optimal control problem .



  min v T Ψ x tf , tf +



tf

 ∗ ∗ T F x ,u ,t udt

(2.127)

t0

subject to .

dx = f (x, u, t) . dt u ∈ U.

x (t0 ) = x0

(2.128) (2.129) (2.130)

r where .x ∗ = x(u∗ , t) is the optimal state vector   and .v ∈ R is the vector of dual variables for the terminal constraints .Ψ x tf , tf = 0. Care must be taken to correctly understand the meaning of optimal control problem (2.127)–(2.130). In particular, this optimal control problem is a mathematical abstraction and of no use for computation, since its criterion depends on knowledge of the variational inequality solution .u∗ . Nonetheless, it is valuable for deriving necessary conditions for DVI(F, f, .Ψ , U, x.0 , t.0 , t.f ). In particular, the necessary conditions for DVI(F, f, .Ψ , U, x.0 , t.0 , t.f ) follow directly from the minimum principle and related necessary conditions for (2.127)–(2.130). In what follows, we will need the Hamiltonian for (2.127)–(2.130), namely

 T H (x, u, λ, t) = F x ∗ , u∗ , t u + λT f (x, u, t)

.

(2.131)

where .λ (t) is the adjoint vector that solves the adjoint equations and satisfies the transversality conditions for the given state variables and controls. Note that, for a given state vector and a given instant in time, the expression (2.131) is convex in u when DVI(F, f, .Ψ , U, x.0 , t.0 , t.f ) is regular in the sense of Definition 2.18. It is now a relatively easy matter to derive the necessary conditions stated in the following theorem: Theorem 2.14 (Necessary Conditions for DVI.(F, f, Ψ, U, x0 , t0 , tf )) Consider the differential variational inequality DVI(F, f, .Ψ , U, x.0 , t.0 , t.f ) defined by (2.126)

2.5 Differential Variational Inequalities

65

with .t0 , .x(t0 ), and .tf fixed. When regularity in the sense of Definition 2.18 holds, necessary conditions for .u∗ ∈ U to be a solution are: 1. the variational inequality: .

   T    T  F x ∗ , u∗ , t + ∇u λ∗ f x ∗ , u∗ , t u − u∗ ≥ 0 ∀u ∈ U

(2.132)

2. the state initial-value problem:   dx ∗ = f x ∗ , u∗ , t . dt ∗ x (t0 ) = x0

(2.133)

.

(2.134)

3. the adjoint dynamics: (−1)

.

  T  dλ∗ = ∇x λ∗ f x ∗ , u∗ , t dt

(2.135)

4. the transversality conditions: 

    ∂Ψ x ∗ tf , tf   λ tf = v T ∂x tf

.

Proof The Pontryagin minimum principle is a necessary condition for optimal control problem (2.127) through (2.130). Hence     u∗ = arg min H x ∗ , u, λ∗ , t

.

u∈U

(2.136)



for each .t ∈ t0 , tf , which in turn has the necessary condition .

  T  ∇u H x ∗ , u∗ , λ∗ , t u − u∗ ≥ 0

u, u∗ ∈ U

Note that   ∇u H (x, u, λ, t) = F x ∗ , u∗ , t + ∇u λT f (x, u, t)

.

where for given u



    , t ∂Ψ x t dλ f f   = ∇x H (x, u, λ, t) , λ tf = v T .λ (u, t) = arg (−1) dt ∂x tf 

 T dλ u + ∇x λT f (x, u, t) , = arg (−1) = ∇x F x ∗ , u∗ , t dt

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2 Mathematical Preliminaries





    T ∂Ψ x tf , tf λ tf = v ∂x tf

     ∂Ψ x t , t dλ f f   = ∇x λT f (x, u, t) , λ tf = v T = arg (−1) dt ∂x tf (2.137) since .x(u, t) is completely determined by knowledge of the controls u. The theorem follows immediately. 

2.6 Nash Games and Differential Nash Games A mathematical game is a mathematical representation of some form of competition among agents or “players” of the game. Most mathematical games have rules of play, agent-specific utilities or payoffs, and a notion of solution. These may be expressed in two fundamental ways: the so-called extensive form and the normal form. A game in extensive form is a presentation, usually via a table or a decision tree, of all possible sequences of decisions that can be made by the game’s players. This presentation is, by its very nature, exhaustive and potentially tedious or even impossible for large games involving multiple players and numerous decisions. By contrast a game in normal form is expressed via mappings, equations, inequalities, and extremal principles. As such, large normal form games are potentially much more computationally tractable, since they may draw upon the computational methods of mathematical programming and optimal control theory, as well as general variational methods.

2.6.1 Nash Equilibria and Normal Form Games The best understood and most widely used mathematical games are noncooperative games, wherein game players, also called agents, act selfishly. A noncooperative mathematical game in normal form uses equations, inequalities, and extremal principles to describe competition among agents—who are intrinsically in conflict and do not collude—informed by some notion of utility and acting according to rules known by the agents of the game. We are especially interested in a notion of solution of noncooperative games known as a Nash equilibrium (named after John Forbes Nash, who proposed it). A set of actions undertaken by the noncooperative agents of interest is a Nash equilibrium if each agent knows the equilibrium strategies of the other agents, and no agent has anything to gain by unilaterally changing his/her own strategy. In particular, if no agent can benefit by changing his/her

2.6 Nash Games and Differential Nash Games

67

strategy while the other agents keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium. As such, finding the Nash equilibrium of a noncooperative game in normal form is not generally equivalent to a single optimization problem but is, rather, naturally articulated as a family of coupled optimization problems. We will learn how, for certain assumptions, those coupled optimization problems may be expressed as a variational inequality. Certain nonextremal problems have a structure that makes them quite amenable to analysis and solution. For our purposes in this chapter, the nonextremal problems known as fixed-point problems, variational inequality problems, and nonlinear complementarity problems are the most important; below, we define each in turn. The following definition will apply: Definition 2.19 (Nash Equilibrium) Suppose there are N agents, each of which chooses a feasible strategy vector .x i from the strategy set .i that is independent of the other players’ strategies. Furthermore, every agent .i ∈ {1, . . . , N } has a cost (disutility) function .i (x) :  −→ R1 that depends on all agents’ strategies where =

N &

.

i

i=1

  x = x i : i = 1, . . . , N Every agent .i ∈ [1, N] seeks to solve the problem .

min i (x i , x −i ) s.t. x i ∈ i

(2.138)

for each fixed yet arbitrary non-own tuple   x −i = x j : j = i

.

A Nash equilibrium is a tuple of strategies x, one for each agent, such that each .x i solves the mathematical program (2.138) and is denoted as .N E(, ). In other words no agent may lower his/her cost (disutility) by unilaterally altering his/her strategy. When the strategy set of any agent .i ∈ [1, N] depends on non-own strategies .x j where .j = i, extension of the definition of a Nash equilibrium is called a generalized Nash equilibrium. That is, we have the following definition: Definition 2.20 (Generalized Nash Equilibrium) Suppose there are N agents, each of which chooses a feasible strategy vector .x i from the strategy set .i (x)

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2 Mathematical Preliminaries

that depends on the strategies of all agents where   x = x i : i = 1, . . . , N

.

Furthermore, every agent .i ∈ {1, . . . , N} has a cost (disutility) function .i (x) :  (x) −→ R1 that depends on all agents’ strategies where  (x) =

N &

.

i (x)

i=1

Every agent .i ∈ [1, N] seeks to solve the problem .

min i (x i , x −i ) s.t. x i ∈ i (x)

(2.139)

for each fixed yet arbitrary non-own tuple   x −i = x j : j = i

.

A generalized Nash equilibrium is a tuple of strategies x, one for each agent, such that each .x i solves the mathematical program (2.139), and is denoted as .GNE(, ).

2.6.2 Differential Nash Games and Differential Nash Equilibria In this section, we want to develop definitions and formulations of dynamic games that employ generalizations of the notions of Nash and generalized Nash equilibria, as solution concepts. We will be solely concerned with open-loop games. An openloop game is one for which initial information is perfect and complete solution trajectories from the start time .t0 to the end time .tf can be calculated, without reliance on any feedback. We need to stipulate that each agent .i ∈ {1, . . . , N} has its own state and own control tuples, namely, .x i and .ui ∈ i , where .i is the set of admissible controls for agent .i ∈ {1, . . . , N}. The non-own control and non-own state vectors faced by agent .i ∈ {1, . . . , N} are .u−i and .x −i where u=

.

ui u−i

! x=

xi x −i

!

for each partition of variables into own and non-own tuples. We will employ the following definition of a differential Nash equilibrium:

2.6 Nash Games and Differential Nash Games

69

Definition 2.21 (Differential Nash Equilibrium) Suppose there are N agents, each of which chooses a feasible strategy vector .ui from the strategy set .i that is independent of the other players’ strategies. Furthermore, every agent .i ∈ {1, . . . , N} has a cost (disutility) functional .Ji (u) :  −→ R1 that depends on all agents’ strategies where =

N &

.

i

i=1

  u = ui : i = 1, . . . , N Every agent .i ∈ [1, N] seeks to solve the problem .

     min Ji (ui , u−i ) = Ki x i tf , tf +

tf

i (x i , ui , x −i , u−i , t)dt

(2.140)

t0

subject to .

  dx i = f i x i , ui , t . dt

x i (t0 ) = x0i .     Ψ i x i tf , tf = 0. ui ∈ i ,

(2.141) (2.142) (2.143) (2.144)

for each fixed yet arbitrary non-own control tuple   u−i = uj : j = i

.

where .x0i is a vector of initial values of .x i , the state tuple of the ith agent, and   x −i = x j : j = i

.

is the corresponding non-own state tuple. A differential Nash equilibrium is a tuple of strategies u such that each .ui solves the optimal control problem (2.140)–(2.144); that equilibrium is denoted as .DN E(, f, K, , , x0 , t0 , tf ). In other words, we again have the situation wherein no agent may lower his/her cost (disutility) by unilaterally altering his/her strategy. The vectors and mappings

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2 Mathematical Preliminaries

intrinsic to Definition 2.21 are the following:  mi ui ∈ L2 t0 , tf  ni x i ∈ H 1 t0 , tf

.

x0i ∈ Rni m = m 1 + m 2 + . . . + mN n = n1 + n2 + . . . + nN  n  2 m

 i : H 1 t0 , tf × L t0 , tf × R1+ −→ L2 t0 , tf   ni  2 mi ni f i : H 1 t0 , tf × L t0 , tf × R1+ −→ L2 t0 , tf K i : Rni × R1+ −→ R1 Ψ i : Rni × R1+ −→ Rri for each agent .i ∈ [1, N]. It is intuitive that a differential Nash equilibrium may be represented as a differential variational inequality. In fact, the following result holds: Theorem 2.15 (Differential Variational Inequality Equivalent to Differential Nash Equilibrium) Take .t0 , .x(t0 ), and .tf to be fixed. There is a differential variational inequality equivalent to the Nash equilibrium DNE(., f,    differential  K, .Ψ , U , .x0 , t.0 , t.f) when .f i x i , ui , t and .i x i , ui , x −i , u−i , t are convex and   continuously differentiable with respect to . x i , ui for all fixed non-own tuples  −i −i  . x , u , for all .i ∈ [1, N]. Proof The relevant Hamiltonian for each agent .i ∈ [1, N] is   T  Hi (x i , ui , λi , t; x −i , u−i ) = i (x i , ui , x −i , u−i , t) + λi f i x i , ui , t

.

and the minimum principle provides, by virtue of convexity, the necessary and sufficient condition .

  T  ∇ui Hi (x i , ui , λi , t; x −i , u−i ) v i − ui ≥ 0 for all v i ∈ i

(2.145)

2.6 Nash Games and Differential Nash Games

71

where the .λi are tuples of adjoint variables determined by  T dλi = ∇x i λi f i (x, u, t) dt

     ∂ i x tf , tf i   λ tf = ∂x tf

  

    T

    i x tf , tf = K i x tf , tf + γ i Ψ i x tf , tf .

(−1)

for agent .i ∈ [1, N]. Let us define the tuples xi .y = λi ⎛ i

! and y

−i

x −i λ−i

=

!

⎞ fi ⎠ gi = ⎝  i T i (−1)∇x i λ f (x, u, t)

⎞

    i x tf , tf ⎜

  

  ⎟ ⎟ i i x tf , tf = ⎜ ⎝ i   ∂ x tf , tf ⎠ = 0   λ tf − ∂x tf ⎛

for each .i ∈ [1, N], so that   y = y i : i = 1, . . . , N   g = g i : i = 1, . . . , N    = i : i = 1, . . . , N .

Also y (t0 ) = y0 =

.

x (t0 ) λ (t0 ) free

!

In addition we define .

Gi (y i , ui , t; y −i , u−i ) = ∇ui Hi (x i , ui , λi , t; x −i , u−i ) .   G = Gi : i = 1, . . . , N

(2.146) (2.147)

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2 Mathematical Preliminaries

It follows from (2.145) and the above notation that

.



tf

⎫ u∗ ∈  ≡ N ⎬ i=1 i

 ∗  ∗ T   ∗ G(y u , t , u , t) v − u dt ≥ 0 ∀v ∈  ⎭

(2.148)

t0

where  

  dy .y(u, t) = arg = g (y, u, t) , y (t0 ) = y0 ,  y tf , tf = 0 dt 

(2.149)

If given differential variational inequality (2.148) and (2.149), by selecting .v j = u∗j for all .j = i, the minimum principle is recovered for each individual .i ∈ {1, . . . , N}. 

2.6.3 Generalized Differential Nash Equilibria When the strategy set and dynamics of any agent .i ∈ [1, N] depend on non-own strategies .u−i and non-own states .x −i , extension of the definition of a differential Nash equilibrium to a generalized differential Nash equilibrium is exactly what we would expect. That is, we have the following definition: Definition 2.22 (Generalized Differential Nash Equilibrium) Suppose there are N agents, each of which chooses a feasible strategy vector .ui from the strategy set .i (u) that depends on all agents’ strategies where   u = ui : i = 1, . . . , N

.

Furthermore, every agent .i ∈ [1, N] has a cost (disutility) functional .Ji (u) :  (u) −→ R1 that depends on all agents’ strategies where  (u) =

N &

.

i (u)

i=1

Every agent .i ∈ [1, N] seeks to solve the problem      i . min Ji (u , u ) = Ki x t f , t f + i

−i

tf

t0

i (x, u, t)dt

(2.150)

2.7 The Scalar Conservation Law

73

subject to .

dx i = f i (x, u, t) . dt

x i (t0 ) = x0i .

   Ψ i x tf , tf = 0. ui ∈ i (u) ,

(2.151) (2.152) (2.153) (2.154)

for each fixed yet arbitrary non-own control tuple   u−i = uj : j = i

.

where .x0i is a vector of initial values of .x i , the state tuple of the ith agent, and   x −i = x j : j = i

.

is the corresponding non-own state tuple. A generalized differential Nash equilibrium is a tuple of strategies u such that each .ui solves the optimal control problem (2.150)–(2.154) and is denoted as .GDNE(, f, K, , U, x0 , t0 , tf ). It is straightforward to define the notion of a differential quasivariational inequality; in turn it is possible to show that .GDNE(, f, K, , U, x0 , t0 , tf ) is equivalent to a differential quasivariational inequality. Of course, a generalized differential Nash equilibrium may be represented as a differential quasivariational inequality, as the reader may easily verify.

2.7 The Scalar Conservation Law In the fields of traffic flow theory and modeling, a large class of fluid-based flow models are represented by one type of partial differential equation (PDE) known as the scalar conservation law (SCL). The SCL is a simple way of describing the temporal-spatial evolution of vehicle density based on conservation of fluid particles. Such a PDE model was first introduced to traffic analysis by Lighthill and Whitham (1955) and Richards (1956).

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2 Mathematical Preliminaries

2.7.1 Definition and Examples of the Scalar Conservation Law A scalar conservation law in a single space dimension is a first-order partial differential equation of the form .

∂ ∂ u(t, x) + f (u) = 0 ∂t ∂x

(2.155)

where .t ∈ R denotes time and .x ∈ R is the space variable; .u(t, x) ∈ R is the solution to (2.155); .f (·) : R → R is called the flux function or, in the traffic modeling context, the fundamental diagram. Equations of the form (2.155) are often used to describe transport phenomena. The unknown .u(t, x) is also called the conserved quantity since (2.155) can be integrated over a given spatial interval .[a, b] to read

.

d dt



b

 u(t, x) dx =

a

a

b

∂ u(t, x) dx = − ∂t



b

a

∂ f (u(t, x)) dx ∂x

= f (u(t, a)) − f (u(t, b))

(2.156)

This means that the total volume contained inside any interval .[a, b] can change only due to the flow of u across the boundaries a and b. Definition 2.23 (Weak Solution) A function .u(t, x) is called a weak solution of (2.155) if .

    ∂ ∂ u · φ + f (u) · φ dx dt = 0 ∂t ∂x

(2.157)

for every continuously differentiable function with compact support. Example 2.6 (The Lighthill-Whitham-Richards Model) We denote the density of vehicles on a highway by .ρ(t, x), at a given location x and time t. As a firstorder approximation, assume that the average vehicle speed .v(t, x) around a point in the temporal-spatial domain is a function of the local density .ρ(t, x); that is,   v(t, x) = v ρ(t, x)

.

As a consequence, the flow of vehicles at .(t, x) is given by     f = f ρ(t, x) = ρ(t, x) · v ρ(t, x)

.

The function .ρ → ρ · v(ρ) is often known as the fundamental diagram. According to the conservation of vehicles, the temporal-spatial evolution of vehicle density is

2.7 The Scalar Conservation Law

75

described by a scalar conservation law of the form .

∂ ∂ ρ(t, x) + f (ρ(t, x)) = 0 ∂t ∂x

(t, x) ∈ [0, T ] × [0, L]

(2.158)

where .T > 0 and the road segment of interest is expressed as a spatial interval [0, L]. Some choices of the fundamental diagram include the following:

.

(i) The Greenshields fundamental diagram (Greenshields, 1935). In this model, the relationship between local density and local speed is affine. That is v(ρ) = vmax 1 −

.

!

ρ

ρ ∈ [0, ρj am ]

ρj am

where .vmax denotes the maximum (free-flow) speed, while .ρj am denotes the maximum (jam) density. With such a choice for the velocity function, the flux function is quadratic in .ρ: f (ρ) = vmax 1 −

.

ρ ρj am

! ·ρ

ρ ∈ [0, ρj am ]

The Greenshields fundamental diagram is illustrated in Fig. 2.1. Notice that there exists a unique .ρ ∗ ∈ [0, ρj am ] where .f (·) attains its maximum. (ii) A series of related studies (Daganzo, 1994, 1995; Newell, 1993a,b,c) presented a simplified kinematic wave model based on a triangular fundamental diagram. Such a choice of the flow-density relation, when combined with a variational approach, yields a simplified representation of the system dynamics. Daganzo (1994) and Daganzo (1995) later propose the cell transmission model, which is viewed as the discrete counterpart of the continuous kinematic wave model with a triangular/trapezoidal fundamental diagram. v(ρ )

f(ρ )

v max

f max

ρmax

ρ

ρ∗

ρmax

ρ

Fig. 2.1 The Greenshields fundamental diagram. Left: the affine relationship between density .ρ and velocity .v(ρ). Right: the quadratic fundamental diagram (Greenshields)

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2 Mathematical Preliminaries

v (ρ )

f (ρ )

v max

f max

ρmax

ρ∗

ρ

ρ∗

ρmax

ρ

Fig. 2.2 The Daganzo-Newell fundamental diagram. Left: the piecewise defined relationship between .ρ and velocity .v(ρ). Right: the triangular fundamental diagram (Daganzo-Newell)

The Daganzo-Newell triangular fundamental diagram stipulates that f (ρ) =

.

vmax ρ ρ ∈ [0, ρ ∗ ] ,   −w ρ − ρj am ρ ∈ (ρ ∗ , ρj am ]

(2.159)

where w is the unique speed of the backward propagating kinematic wave. As a consequence of (2.159), the following velocity-density relationship exists in the Daganzo-Newell fundamental diagram (see Fig. 2.2 for an illustration of the Daganzo-Newell fundamental diagram): ⎧ ⎨vmax

∗ ! ρ ∈ [0, ρ ] .v(ρ) = ρ ⎩−w 1 − j am ρ ∈ (ρ ∗ , ρj am ] ρ

(2.160)

2.7.2 Characteristics, Shock Waves, and Weak Solutions Consider the conservation law with initial condition: ⎧ ⎨ ∂ u(t, x) + ∂ f ρ(t, x) = 0 (t, x) ∈ (0, ∞) × R . ∂t ∂x ⎩u(0, x) = g(x) x∈R

(2.161)

where .g(·) is a locally integrable function defined on .R. In addition, assume that f (·) is continuously differentiable. Consider the parametrized curve .(t, x(t)) ⊂ R2

.

2.7 The Scalar Conservation Law

77

given by the following ordinary differential equation

.

  x(t) ˙ = f  u(t, x) x(0) = x0

(2.162)

where .x(·) ˙ denotes the time derivative of .x(·). If .u(t, x) is the solution to (2.161), then one can verify that .

   ∂  ∂  d  u t, x(t) = u t, x(t) + u t, x(t) · x(t) ˙ dt ∂t ∂x    ∂  ∂ = u(t, x(t)) + f  u(t, x) · u t, x(t) = 0 ∂t ∂x

In  otherwords, if .u(t, x) is a solution, then it attains the same value along the curve t, x(t) defined via the ODE in (2.162). Moreover, such a value can be obtained from the initial condition:

.

    u t, x(t) = u 0, x(0) = g(x0 )

.

The curves .t → x(t) are called characteristics. Notice that in the case of scalar conservation law (2.161), the characteristics are line segments since     x(t) ˙ = f  g(x0 ) ⇒ x(t) = x0 + t · f  g(x0 )

.

(2.163)

Example 2.7 (Characteristic Lines with Initial Condition) Let us consider the initial-value problem for the LWR model with the Greenshields fundamental diagram: ⎧   ⎨ ∂ ρ + ∂ ρ 1 − ρ = 0 (t, x) ∈ (0, +∞) × R . ∂t ∂x ⎩ρ(0, x) = ρ (x) x∈R 0

(2.164)

Here we assume for simplicity that both .vmax and .ρj am are 1. The initial condition ρ0 (·) is given by

.

ρ0 (x) =

.

⎧ ⎪ ⎪ ⎨1/4

x ∈ (−∞, −1)

3/4 ⎪ ⎪ ⎩0

x ∈ (1, +∞)

x ∈ [−1, 1]

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2 Mathematical Preliminaries

According to (2.163), the characteristic lines are locally defined to be the straight lines of the following form    t, x0 + t · − 2ρ0 (x0 ) + 1

t →

.

The implication of the above statement is that, given any point .(t, x), if there exists a unique characteristic line with slope .s = dx/dt that passes through .(t, x), then the solution at this point becomes ρ(t, x) = ρ0 x −

.

t s

!

A graphical illustration of this principle is presented in Fig. 2.3. Given each point in region I, III, or V, a unique characteristic line passes through it. Therefore, the solution at this point can be determined by moving along the characteristic line toward the domain .{t = 0} × R where the initial value is specified. Notice that in Fig. 2.3, the characteristic lines emitting from .(−∞, −1) and .[−1, 1] intersect each other after a certain time. As a consequence, given any point in region IV, there are two characteristic lines that pass through it. In this case the solution is no longer well-defined by the method of characteristics; and the concept of shock waves needs to be introduced to resolve the conflicting characteristic lines. A shock wave is a jump discontinuity (in space) in the solution .ρ(·, ·) that propagates in the .t − x domain. It resolves the conflict among characteristics in a mathematically precise manner; see Proposition 2.2 below. One important observation from the previous example is that a solution of a scalar conservation law may produce a discontinuity (shock wave) in finite time and become discontinuous. This is true even if the initial condition .g(·) is smooth.

I

II

IV

III

V

t

ρ0( x ) = 1/4

−1

ρ0( x ) = 3/4

1

ρ0( x ) = 0

x

Fig. 2.3 Illustration of the method of characteristics for solving scalar conservation laws with initial value

2.7 The Scalar Conservation Law

79

To achieve a global existence result, it is essential to work within a class of discontinuous functions, interpreting the equations of the form (2.155) in their distributional sense (2.157). Definition 2.24 (Weak Solution of the Initial-Value Problem) Let .f (·) : R → R be a continuous function. Fix any locally integrable function .g(·) : R → R and .T > 0. A function .ρ : [0, T ] × R → R is a weak solution to the following initialvalue problem (Cauchy problem) ⎧ ⎨ ∂ ρ(t, x) + ∂ f ρ(t, x) = 0 . ∂t ∂x ⎩ρ(0, x) = g(x) x∈R

(t, x) ∈ (0, T ) × R

(2.165)

if for every smooth test function .φ : [0, T ] × R → R with compact support, the following holds 

T



+∞

.

−∞

0

 +

ρ(t, x)

+∞ −∞

  ∂ ∂ φ(t, x) + f ρ(t, x) φ(t, x) dx dt ∂t ∂x

g(x) φ(0, x) dx = 0

(2.166)

Note that the definition of a weak solution is obtained by temporarily assuming that ρ is smooth and by rewriting the scalar conservation law into a form of (2.166). The idea is to multiply the conservation law by a smooth function .φ and then to integrate by parts. As an immediate consequence of the definition above, the following result, known as the Rankine-Hugoniot condition resolves the conflict of intersecting characteristics by providing a characterization of the shock wave.

.

Proposition 2.2 (The Rankine-Hugoniot Condition) Suppose in some region Ω ⊂ [0, T ] × R that the weak solution .ρ of (2.165) is smooth on either side of a curve .C. In addition, suppose .C can be represented parametrically as .{(t, x) : x = s(t)} for some function .s(·) : [0, T ] → R. Fix .(t0 , x0 ) ∈ C and let the values of .ρ on the left and right sides of C be denoted by .ρl and .ρr , respectively. Then .

s˙ (t0 ) · (ρl − ρr ) = f (ρl ) − f (ρr )

.

(2.167)

where .s˙ (t0 ) is recognized as the speed of the curve .C at .t0 . Proof See Chapter 3 of Evans (1995) for a detailed proof.



Equation (2.167) yields the well-known formula for calculating the speed of a shock given the two distinct density values on both sides.

80

2 Mathematical Preliminaries

2.7.3 Non-uniqueness of Integral Solutions, Entropy Conditions To motivate our discussion of entropy conditions, let us return to Example 2.7 and in particular, Fig. 2.3. While the conflicting characteristics can be resolved by the definition of weak (integral) solutions and the Rankine-Hugoniot condition (shock wave), the method of characteristics does fail to provide any information within region II. To highlight this issue, let us consider the following local solution near region II. ρ1 (t, x) =

x−1 t x−1 t

3/4

.

0

< 1/4 > 1/4

(2.168)

Figure 2.4 left illustrates .ρ1 (t, x). It is not difficult to verify that the RankineHugoniot condition holds for .ρ1 (·, ·): ul = 3/4, ur = 0 ⇒ s˙ =

.

1 dx = dt 4

Thus, .ρ1 (·, ·) as defined by (2.168) is a weak solution in the sense of Definition 2.24. Now, consider another locally defined solution

ρ2 (t, x) =

.

⎧ ⎪ ⎪ ⎨3/4

t−x+1 ⎪ 2t

⎪ ⎩0

x−1 t

< − 1/2

−1/2 < x−1 t

x−1 t

< 1

(2.169)

> 1

Figure 2.4 right illustrates .ρ2 (t, x). The function .ρ2 (·, ·) is said to contain a rarefaction wave. It is also a weak solution. The example above suggests that the weak solution in the sense of Definition 2.24 is in general not unique. In fact, .ρ1 (·, ·) defined in (2.168) is a nonphysical solution. To exclude this and various other nonphysical solutions, an additional condition needs to be introduced. One such condition is called the entropy condition. The entropy condition can be defined in various ways; one example is the Kruzkov entropy condition (Bressan, 2000). In this book, we offer a somewhat simpler and intuitive notion of an entropy condition for scalar conservation laws of the form (2.161). Moving forward in time, one may encounter the conflict of characteristic lines, a circumstance that results in solution discontinuities. Nevertheless, when starting at some point in the interior of the domain and moving backward in time along a characteristic, one should expect not to cross any other characteristic lines. With this insight, we return to the general initial-value problem (2.161). Suppose at some point on a curve s of discontinuity that the solution has distinct left and right limits, .ul and .ur , and that two characteristics, from the left and the right,

2.7 The Scalar Conservation Law

ρ=3/4

81

ρ=0

ρ=3/4

t

ρ=0

t

x

1

1

x

Fig. 2.4 Illustration of the non-uniqueness of weak solutions. Left: .ρ1 (·, ·) of (2.168) is not a physical (entropy) solution. Right: .ρ2 (·, ·) of (2.169) with a rarefaction wave is an entropy solution

respectively, intercept the curve s at this point. Then, since the slopes of these two intersecting characteristics are given by f  (ul ) and f  (ur ),

.

respectively, where f is the flux function in (2.161). Then there must hold f  (ul ) > s˙ > f  (ur )

(2.170)

.

Inequalities (2.170) are called the entropy conditions. A weak solution to (2.161) that satisfies (2.170) is called a weak entropy solution. In the previous example, .ρ1 defined in (2.168) is not an entropy solution since the characteristic lines are intersecting each other when moving backward in time (Fig. 2.4). On the other hand, .ρ2 defined in (2.169) contains a rarefaction wave and is a continuous entropy solution. Example 2.8 (Characteristic Lines Continued) We continue Example 2.7 to find the weak entropy solution on the whole half plane .(t, x) ∈ R+ × R. As shown in Fig. 2.3, the conflict among characteristic lines emitting from .(−∞, −1) and .(−1, 1) is resolved by the shock wave .s(·) with speed given by the RankineHugoniot condition:

s˙ =

.

1 4

  1 − 14 − 1 4

−

3 4 3 4

  1 − 34

= 0

82

2 Mathematical Preliminaries

Thus the solution for .0 ≤ t ≤ 4 is ⎧ ⎪ 1/4 ⎪ ⎪ ⎪ ⎨3/4 .ρ(t, x) = t−x+1 ⎪ ⎪ 2t ⎪ ⎪ ⎩ 0

x ≤ −1 −1 < x ≤ − − 2t

t 2

+1

0 ≤ t ≤ 4

+1 < x ≤ t +1

x > t +1

When .t > 4, the shock located at .x = −1 and .t = 4 continues to propagate; but the right limit of the solution .ρr is no longer a constant .3/4. Instead, .ρr is given by the rarefaction wave: ρr (t, x) =

.

t −x+1 2t

We can then set up an ordinary differential equation for the trajectory of the shock wave. Namely, the Rankine-Hugoniot condition dictates that

.

dx = dt

  1 − 14 −



t−x+1 1 − t−x+1 2t 2t 1 t−x+1 4 − 2t

1 4

 (2.171)

The above ODE is associated with the following value condition: x(4) = − 1

.

(2.172)

The ODE (2.171)–(2.172) can be solved analytically: x(t) =

.

√ t −2 t +1 2

Therefore, the solution for .t > 4 reads ⎧ √ ⎪ x ≤ 2t − 2 t + 1 ⎪ ⎨1/4 √ t t−x+1 .ρ(t, x) = 2t 2 −2 t +1 < x ≤ t +1 ⎪ ⎪ ⎩0 x > t +1 The weak entropy solution is presented graphically in Fig. 2.5.

t > 4

2.8 The Hamilton-Jacobi Equations and the Variational Principle

83

t 1

1/2

0 −1

x

1

Fig. 2.5 The weak entropy solution .ρ(t, x) of (2.164) on the half plane .(t, x) ∈ R+ × R

2.8 The Hamilton-Jacobi Equations and the Variational Principle A scalar conservation law of the form (2.155) is closely related to a Hamilton-Jacobi equation, as we shall demonstrate in this section.

2.8.1 The Hamilton-Jacobi Equation Let .ρ(t, x) and .f (ρ(t, x)) be the vehicle density and flow at location x and time t, respectively. We introduce the following Moskowitz functions (Moskowitz, 1965) .N (t, x), also known as N-curves (Newell, 1993a,b,c), which satisfy  N (t2 , x2 ) − N(t1 , x1 ) = −

x2

.

 −ρ(t1 , x) dx +

x1

∀t1 , t2 ∈ [0, T ],

t2

  f ρ(t, x2 ) dt

t1

∀x1 , x2 ∈ [a, b]

(2.173)

84

2 Mathematical Preliminaries

Note that .N(t, x) measures the cumulative number of vehicles that have passed location x by time t. As a consequence of definition (2.173), the following identities hold: .

  ∂ N(t, x) = f ρ(t, x) ∂t

∂ N(t, x) = − ρ(t, x) ∂x

(2.174)

for almost every t. The reason that (2.174) may not hold for every t is that .N (t, x) is not continuously differentiable in general. It is a simple exercise to show conceptually that .N(t, x) solves the following Hamilton-Jacobi equation. .

  ∂ ∂ N(t, x) − f − N(t, x) = 0 ∂t ∂x

(2.175)

However, the exact relationship between the scalar conservation law (2.155) and the H-J equation (2.175) needs to be established by articulating in which sense a solution is defined, which is omitted here for the sake of brevity. The reader is referred to Evans (1995) for a rigorous mathematical treatment.

2.8.2 The Variational Theory The notion of a variational approach for solving the Hamilton-Jacobi equation was initially derived from calculus of variations (Evans, 1995; Lax, 1957) and then from viability theory (Aubin et al., 2008; Claudel & Bayen, 2010a,b). It has also been discussed in the context of modeling homogeneous road traffic on a single road segment (Daganzo, 2005, 2006; Newell, 1993a,b,c).

2.8.2.1

The Classical Lax-Hopf Formula

The H-J equation (2.175) is a special case of the more general Hamilton-Jacobi equation with an initial condition: .

∂ ∂ N(t, x) + H N(t, x) ∂t ∂x

! = 0

N(0, x) = g(x)

(t, x) ∈ [0, ∞) × R. x∈R

(2.176) (2.177)

There are three main categories of variational methods for the H-J equation. Each is usually referred to as a formula, and the categories of such formulas are Lax-Hopf, Lax-Oleinik, and generalized Lax-Hopf. The selection of variational method is based on solution class and type of initial/boundary conditions of interest. The following

2.8 The Hamilton-Jacobi Equations and the Variational Principle

85

theorem establishes a semi-analytical representation for viscosity solutions; see Evans (1995) for a precise definition of viscosity solutions. Theorem 2.16 (Lax-Hopf Formula) Suppose the Hamiltonian .H is continuous and convex, .g(·) : R → R is Lipschitz continuous, then   x − y  + g(y) N(t, x) = inf tL y∈R t

.

(2.178)

is the unique viscosity solution to the initial-value problem (2.176)–(2.177) where L is the Legendre transformation of .H:

.

  L(q) = inf H(p) − qp

.

p

Proof We refer the reader to Evans (1995).

(2.179) 

The solution given by (2.178) is continuous and satisfies the initial condition at almost every point x. However, such a formula can only accommodate initial conditions. For an arbitrary link in a general traffic network, the relevant scalar conservation law (or the Hamilton-Jacobi equation) is normally constrained by not only the initial condition, but also upstream and downstream boundary conditions and, in some cases, internal boundary conditions (Claudel & Bayen, 2010a,b). To accommodate such needs, and to extend the solution class to include possibly discontinuous functions, Aubin et al. (2008) propose a class of semi-lower continuous solutions to the H-J equation, using viability theory. This solution class, called the viability episolution, is of the Barron-Jensen/Frankowska type (Barron & Jensen, 1990; Frankowska, 1993); and it is given by the generalized Lax-Hopf formula.

2.8.2.2

The Generalized Lax-Hopf Formula

It is most convenient to illustrate the notion of viability episolution and the generalized Lax-Hopf formula by looking at a homogeneous road segment. Let us fix a spatial-temporal domain .[0, T ] × [a, b] where .[0, T ] is the time horizon, .b − a = L is the length of the link. The articulation of the generalized Lax-Hopf formula requires the following definition of value conditions. Definition 2.25 A value condition .C(·, ·) is a lower-semicontinuous function that maps .Ω, a subset of .[a, b] × [0, T ], to .R. In a network extension of the scalar conservation law model, the PDE is often subject to initial, upstream boundary, and downstream boundary conditions. In this case, the domain of the value condition for the link can be expressed as Ω = {0} × [a, b] ∪ [0, T ] × {a} ∪ [0, T ] × {b}

.

86

2 Mathematical Preliminaries

One may extend the domain .Ω of a value condition to .[0, T ] × [a, b] by assigning C(t, x) = +∞ if .(t, x) ∈ / Ω. Such extension is closely related to the inf-morphism property of viability solutions; the reader is referred to Claudel and Bayen (2010a), Claudel and Bayen (2010b) for details. Consider the following scalar conservation law

.

.

 ∂  ∂ ρ(t, x) + f ρ(t, x) = 0 ∂t ∂x

(t, x) ∈ [0, T ] × [a, b]

(2.180)

where .ρ(t, x) denotes local density of cars at location x and time t. It is natural to assume that .ρ(t, x) ∈ [0, ρj ] where .ρj is the jam density corresponding to a bumper-to-bumper situation. The corresponding Hamilton-Jacobi equation is .

  ∂ ∂ N(t, x) − f − N(t, x) = 0 ∂t ∂x

(2.181)

Let us define the concave transformation of the flux function .f (·): f ∗ (u) =

sup

.

ρ∈[0, ρj ]



f (ρ) − uρ



The following generalized Lax-Hopf formula is attributed to Aubin et al. (2008) based on viability theory. Theorem 2.17 (Generalized Lax-Hopf Formula) The viability episolution to (2.181) associated with value condition .C(·, ·) is given by NC (t, x) =

.



inf

(u, τ )∈Dom(f ∗ )×R

+



C(t − τ, x − τ u) + τf ∗ (u)

(2.182)

Next, we will illustrate (2.182) in a more computable form such that it can be used for dynamic network loading as we discuss in Chap. 8. We consider a fundamental diagram .f (·) that is continuous and concave with .f  (0+) = v and  .f (ρj −) = w; see Fig. 2.6a. In addition, we consider the H-J equation with initial, upstream, and downstream conditions as follows: .

∂ ∂t N(t,

 ∂  x) − f − ∂x N(t, x) = 0 (t, x) ∈ [0, T ] × [a, b]

N (0, x) = N ini (x),

N (t, a) = N up (t),

N (t, b) = N dn (t)

(2.183)

Based on the maximum forward and backward wave speeds v and .|w|, we partition the domain .[0, T ] × [a, b] into four disjoint regions .ΩI , .ΩII , .ΩIII , and .ΩIV ; see Fig. 2.6b. Depending on the region in which a given point .(t, x) ∈ [0, T ] × [a, b] lies, the value conditions that influence the solution .N(t, x) vary; see Fig. 2.7. Instantiating the generalized Lax-Hopf formula yields a further elaborated solution as follows:

2.8 The Hamilton-Jacobi Equations and the Variational Principle

87

x

f( )

N dn(t)

b v

C

t

v

w N ini(x)

w a

0

j

c

0

T

N up(t) (b)

(a)

t

Fig. 2.6 (a): A generic fundamental diagram; (b): partition of the domain by region of influence x

x N dn(t)

b

t

N dn(t)

b

v

v N ini(x)

N ini(x) w a

0

w

T

N up(t)

a

t

0

N dn(t)

b

T

N up(t)

t

x

x t

N dn(t)

b

v

t

v

N ini(x)

N ini(x) w

a

t

0

w

T

N up(t)

a

t

0

T

N up(t)

t

Fig. 2.7 Domain of influence of the initial and boundary conditions

• .(t, x) ∈ ΩI : N(t, x) = min A(u; t, x)

(2.184)

.

u∈[w, v]

• .(t, x) ∈ ΩII : N (t, x) = min

.

 min

u∈[ x−a t , v]

B(u; t, x),

min u∈[w,

x−a t ]

A(u; t, x)

(2.185)

88

2 Mathematical Preliminaries

• .(t, x) ∈ ΩIII : 

N(t, x) = min

min

.

u∈[ x−b t , v]

min

A(u; t, x),

u∈[w,

x−b t ]

(2.186)

C(u; t, x)

• .(t, x) ∈ ΩIV : N (t, x) =  min min

.

u∈[ x−a t , v]

 B(u; t, x),

min u∈[w,

x−b t ]

C(u; t, x),

min

u∈[ x−b t ,

x−a t ]

A(u; t, x) (2.187)

where A(u; t, x) = N ini (x − ut) + tf ∗ (u) ! x−a x−a ∗ + B(u; t, x) = N up t − f (u) u u ! x−b x−b ∗ dn t− + f (u) C(u; t, x) = N u u

.

and ΩI ΩII . ΩIII ΩIV

= = = =

{(t, x) ∈ [0, T ] × [a, b] : {(t, x) ∈ (0, T ] × [a, b] : {(t, x) ∈ (0, T ] × [a, b] : {(t, x) ∈ (0, T ] × [a, b] :

x x x x

≥ < ≥
>

b + wt} b + wt} b + wt} b + wt}

(2.188)

Notice that in (2.184)–(2.187), the inner “min” operators correspond to the minimization problem associated with a single condition (initial, upstream boundary, or downstream boundary), while the outer “min” operators represent the inf-morphism property (Claudel & Bayen, 2010a); that is, it specifies the dominance of one condition over the others. Example 2.9 (Lax-Hopf Formula for Triangular Fundamental Diagram) We consider a triangular fundamental diagram with forward kinematic wave speed of .v > 0 and backward wave speed of .w < 0. In this case the transformation .f ∗ can be expressed as f ∗ (u) = C − ρc · u

.

u ∈ [w, v]

where C is the flow capacity and .ρc is the critical density (see Fig. 2.6). Furthermore, assuming that the link of interest is initially empty, which is typical in a dynamic

References and Suggested Reading

89

network loading context, the Lax-Hopf formula yields the following simplified solution: N(t, x) ≡ 0

.

N (t, x) = N up t −

.

(t, x) ∈ ΩI ∪ ΩIII x−a v

(2.189)

! (t, x) ∈ ΩII

(2.190)

! !   x−a x−b up dn t− t− , N + ρc (b − x) .N (t, x) = min N v w (t, x) ∈ ΩIV

(2.191)

References and Suggested Reading Arrow, K. J., & Kurz, M. (1970). Public investment, the rate of return, and optimal fiscal policy. Baltimore: The Johns Hopkins University Press. Aubin, J. P., Bayen, A. M., & Saint-Pierre, P. (2008). Dirichlet problems for some Hamilton-Jacobi equations with inequality constraints. SIAM Journal on Control and Optimization, 47(5), 2348– 2380. Barron, E. N., & Jensen, R. (1990). Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Communications in Partial Differential Equations, 15, 1713–1742. Bressan, A. (2000). Hyperbolic systems of conservation laws. The one-dimensional cauchy problem. Oxford. Bressan, A., & Piccoli, B. (2007). An introduction to the mathematical theory of control. Springfield, MO: American Institute of Mathematical Sciences. Claudel, C. G., & Bayen, A. M. (2010a). Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part I: Theory. IEEE Transactions on Automatic Control, 55(5), 1142–1157. Claudel, C. G., & Bayen, A. M. (2010b). Lax-Hopf based incorporation of internal boundary conditions into Hamilton-Jacobi equation. Part II: Computational methods. IEEE Transactions on Automatic Control, 55(5), 1158–1174. Daganzo, C. F. (1994). The cell transmission model. Part I: A simple dynamic representation of highway traffic. Transportation Research Part B, 28(4), 269–287. Daganzo, C. F. (1995). The cell transmission model. Part II: Network traffic. Transportation Research Part B, 29(2), 79–93. Daganzo, C. F. (2005). A variational formulation of kinematic waves: basic theory and complex boundary conditions. Transportation Research Part B, 39(2), 187–196. Daganzo, C. F. (2006). On the variational theory of traffic flow: well-posedness, duality and application. Network and Heterogeneous Media, 1(4), 601–619. Evans, L. C. (1995). Partial differential equations (2nd ed.). Providence, RI: American Mathematical Society. Frankowska, H. (1993). Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM Journal on Control and Optimization, 31(1), 257–272. Friesz, T. L. (2010). Dynamic optimization and differential games. Springer.

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Greenshields, B. D. (1935). A study of traffic capacity. In Proceedings of the 13th Annual Meeting of the Highway Research Board (Vol. 14, pp. 448–477). Lax, P. D. (1957). Hyperbolic systems of conservation laws II. Communications on Pure and Applied Mathematics, 10(4), 537–566. Lighthill, M., & G. Whitham. (1955). On kinematic waves. II. A theory of traffic flow on long crowded roads. Proceedings of the Royal Society of London: Series A, 229, 317–345. Mangasarian, O. L. (1966). Sufficient conditions for the optimal control of nonlinear systems. SIAM Journal of Control, 4, 19–52. Minoux, M. (1986). Mathematical Programming: Theory and Algorithms. Wiley. Moskowitz, K. (1965). Discussion of ‘freeway level of service as influenced by volume and capacity characteristics’ by D.R. Drew and C.J. Keese. Highway Research Record, 99, 43–44. Newell, G. F. (1993a). A simplified theory of kinematic waves in highway traffic. Part I: General theory. Transportation Research Part B, 27(4), 281–287. Newell, G. F. (1993b). A simplified theory of kinematic waves in highway traffic. Part II: Queuing at freeway bottlenecks. Transportation Research Part B, 27(4), 289–303. Newell, G. F. (1993c). A simplified theory of kinematic waves in highway traffic. Part III: Multidestination flows. Transportation Research Part B, 27(4), 305–313. Richards, P. I. (1956). Shockwaves on the highway. Operations Research, 4, 42–51. Royden, H. L., & Fitzpatrick, P. (1988). Real analysis (Vol. 3). Englewood Cliffs, NJ:: Prentice Hall. Rudin, W. (2006). Functional analysis. McGraw-Hill. Seierstad, A., & Sydsæter, K. (1977). Sufficient conditions in optimal control theory. International Economic Review, 18, 367–391. Seierstad, A., & Sydsæter, K. (1999). Optimal control theory with economic applications. Amsterdam: Elsevier. Walter, W. (1988). Ordinary differential equations. Springer.

Chapter 3

The Variational Inequality Formulation of Dynamic User Equilibria

In this chapter, we present an infinite-dimensional variational inequality (VI) formulation of the continuous-time simultaneous route-and-departure-time (SRDT) dynamic user equilibrium. Such a result was first established in Friesz et al. (1993) using measure theoretic arguments under suitable regularity conditions. There are two essential components within the SRDT notion of DUE: (1) the mathematical expression of Nash-like equilibrium conditions and (2) a network performance model, which is, in effect, an embedded dynamic network loading (DNL) problem. The embedded DNL problem captures the relationship among arc entry flow, arc exit flow, arc delay, and path delay for any path departure rate trajectory. One of the advantages of the variational inequality formulation given by Friesz et al. (1993) is that it subsumes almost all SRDT DUE models regardless of the arc dynamics, flow propagation, and arc delay functions employed. In particular, the arc dynamics and flow propagation constraints may be naturally embedded in the so-called effective delay operator that is viewed as a mapping between two infinite-dimensional Hilbert spaces. However, it would be a mistake to think that somehow the VI formulation is an “easier” formulation since the effective delay operators are generally not knowable in closed form; in fact the delay operators may be non-analytic and may need to be derived from an embedded delay model, data combined with some response surface methodology, or data combined with inverse modeling. We will have more to say about the delay operator when we discuss dynamic network loading in Chaps. 7 and 8.

3.1 Notation and Essential Background Before we introduce variational inequality formulations for a variety of dynamic user equilibria, we will first provide definitions and terminology required to describe the notion of dynamic user equilibrium. © Springer Nature Switzerland AG 2022 T. L. Friesz, K. Han, Dynamic Network User Equilibrium, Complex Networks and Dynamic Systems 5, https://doi.org/10.1007/978-3-031-25564-9_3

91

92

3 The Variational Inequality Formulation of Dynamic User Equilibria

The time interval of analysis is a single commuting period or “day” expressed as [t0 , tf ] ⊂ R where .R denotes the set of real numbers. We let .P be the set of all paths employed by travelers. For each path .p ∈ P, we define the path departure rate to be a function of departure time .t ∈ [t0 , tf ],1 and express it as

.

hp (·) : [t0 , tf ] → R+

.

where .R+ denotes the set of non-negative real numbers. Each path departure rate hp (t) is interpreted as a departure-time density, or, more simply, a path flow, measured at the entrance of the first arc of the relevant path. The units of path flows are vehicles per unit time. We next define .h(·) = {hp (·) : p ∈ P} to be a vector of departure rates (path flows). Therefore, .h = h(·) can be viewed as a vector-valued function of t, the departure time.2 We denote the space of square-integrable functions on the real interval .[t0 , tf ] by .L2 [t0 , tf ] and its subset consisting of non-negative functions by .L2+ [t0 , tf ]. We stipulate that each path flow is square integrable; that is

.

hp (·) ∈ L2+ [t0 , tf ],

.

 |P | h(·) ∈ L2+ [t0 , tf ] ,

  |P | |P | where . L2+ [t0 , tf ] is a cone of the .|P|-fold product space . L2 [t0 , tf ] consisting of non-negative vector-valued functions.3 It is widely known in functional analysis that the space .L2 [t0 , tf ] is a Hilbert space with the following inner product:  .

f, g =

tf

f (t) · g(t) dt

∀f, g ∈ L2 [t0 , tf ]

t0

 |P | We note that the .|P|-fold product space . L2 [t0 , tf ] is also a Hilbert space with the inner product defined as follows:

.

f, g =

|P |  

tf

fp (t) · gp (t) dt

 |P | ∀f, g ∈ L2 [t0 , tf ]

p=1 t0

This choice of space and inner product will be used throughout the remainder of this book. It will be seen that the path departure rates .hp (·), p ∈ P, are defined only up to a set of measure zero. With this in mind, let .ν be a Lebesgue measure on .[t0 , tf ],

1 Throughout this book, we employ the convention that expresses a function/functional/operator g of certain argument to be .g(·) and expresses the image of a particular point x to be .g(x). 2 For notational convenience and without causing any confusion, we will sometimes use h instead of .h(·) to denote path flow vectors. 3 A cone in a vector space is a subset that is closed under multiplication by positive scalars.

3.1 Notation and Essential Background

93

and for each measurable set, .S ⊆ [t0 , tf ], let .∀ν (t ∈ S) represent the phrase for ν-almost all .t ∈ S. If .S = [t0 , tf ], then we may at times simply write .∀ν (t). Here, as in all DUE modeling, the single most crucial ingredient is the path delay operator, which maps a given vector of departure rates h to a vector of path travel times. More specifically, we let

.

Dp (t, h)

.

∀t ∈ [t0 , tf ],

∀p ∈ P

be the path travel time of a driver departing at time t and following path p, given the departure rates associated with all the paths in the network, denoted by h. We then define the path delay operator .D(·) by letting .D(h) = {Dp (·, h) : p ∈ P}, which is a vector of time-dependent path travel times .Dp (t, h). Furthermore, .D(·)  |P | is an operator defined on . L2+ [t0 , tf ] , which maps a vector valued function .h(·) to another vector-valued function .{Dp (·, h) : p ∈ P}. More precisely,  |P |  |P | D : L2+ [t0 , tf ] → L2+ [t0 , tf ] .

(3.1)

h(·) = {hp (·), p ∈ P} → D(h) = {Dp (·, h), p ∈ P} |P |  In (3.1), the first expression means that D is a mapping from . L2+ [t0 , tf ] into itself; the second expression means that .D(h) is an image of h. The same notational meanings of .→ and . → will be used throughout this book. The effective delay operator .Ψ is similarly defined, except that the effective path delay contains, in addition to path travel time, also arrival penalties. Thus, the effective path delay is a more general notion of “travel cost” than path delay. The effective delay operator is defined as follows.  |P |  |P | Ψ : L2+ [t0 , tf ] → L2+ [t0 , tf ] .

(3.2)

h(·) = {hp (·), p ∈ P} → Ψ (h) = {Ψp (·, h), p ∈ P} where   Ψp (t, h) = Dp (t, h) + f t + Dp (t, h) − TA

.

∀t ∈ [t0 , tf ],

∀p ∈ P (3.3)

  TA is the desired arrival time, and .TA < tf . The term .f t + Dp (t, h) − TA assesses a non-negative penalty whenever

.

t + Dp (t, h) = TA

.

(3.4)

94

3 The Variational Inequality Formulation of Dynamic User Equilibria

since .t + Dp (t, h) is the clock time at which departing traffic arrives at the destination of path .p ∈ P. Note that, for convenience, .TA is assumed to be independent of destination. However, that assumption is easy to relax, and the consequent generalization of our model is a trivial extension. Remark 3.1 Our formulation requires minimal assumptions on the functional form of the penalty function .f (·) and can thus accommodate very general arrival time windows, arrival preferences, and special circumstances. For example, if early arrivals are encouraged, then the function .f (·) can be selected such that .f (s) is increasing for .s < 0. In practice, the arrival penalty will be asymmetric, assigning a greater penalty to late arrivals than to early arrivals. In general, however, the function .f (·) needs to be carefully calibrated using behavioral assumptions and empirical data. We interpret .Ψp (t, h) to be the cost of travel for drivers departing at time t following path p, given the vector of path departure rates h. We stipulate that each path effective delay Ψp (·, h) : [t0 , tf ] −→ R++

.

∀p ∈ P

is measurable, strictly positive, and square integrable, where .R++ denotes the set of positive real numbers. The notion of strictly positive functions, as is employed throughout this book, refers to measurable functions that are positive almost everywhere. The notation  |P | . Ψ (h) = {Ψp (·, h) : p ∈ P} ∈ L2+ [t0 , tf ]

.

is used to express the complete vector of effective delays. In order to define an appropriate concept of minimum travel costs in the present context, we require the measure-theoretic analog of the infimum of a set of numbers. In particular, for any measurable function .g(·) : [t0 , tf ] → R, the essential infimum of .g(·) on .[t0 , tf ] is given by   . . essinf {g(s)} = sup x ∈ R : ν{s ∈ [t0 , tf ] : g(s) < x} = 0 (3.5) s∈[t0 , tf ]

Note that for each .x > essinf {g(s)} it must be true by definition that s∈[t0 , tf ]

ν{s ∈ [t0 , tf ] : f (s) < x} > 0

.

Let us define the essential infimum of effective travel delays, which depend on the path flows h:   Ψp (t, h) > 0 .vp (h) = essinf ∀p ∈ P (3.6) t∈[t0 , tf ]

  vij (h) = min vp (h)

.

p∈Pij

∀ (i, j ) ∈ W

(3.7)

3.2 The VI Formulation of DUE with Fixed Demand

95

3.2 The VI Formulation of DUE with Fixed Demand We present the variational inequality (VI) formulation for dynamic user equilibrium with fixed travel demands between origin-destination pairs. This formulation was first given by Friesz et al. (1993) and is an exact formulation of dynamic network user equilibrium, where by “exact” we mean a model that is completely mathematically internally consistent and involves no ad hoc treatment of delay operators, departure time choice, flow propagation anomalies, or other critical model features prior to its numerical solution. The Friesz et al. (1993) formulation employs path delay operators that obey appropriate arc dynamics and incorporate a path flow propagation mechanism. This embedded path flow propagation mechanism ensures arc entry and exit at appropriate times along a given path and preserves the first-in-first-out (FIFO) queue discipline when appropriate regularity conditions are met. The requirement that the delay operators reflect arc level dynamics and flow propagation considerations make the delay operators unknowable in closed form. The flow propagation mechanism of the Friesz et al. (1993) formulation depends on arc exit time functions and their inverses. Inverse exit time functions, like the path delay operators, cannot be known in closed form. The Friesz et al. (1993) model expresses a dynamic Nash-like equilibrium relative to departure time and path choice as an infinite-dimensional variational inequality. This variational inequality cannot be solved by traditional methods since it is based on nonanalytic path delay operators which are only known numerically. In subsequent papers, Friesz et al. (2001) and Friesz and Mookherjee (2006) developed a differential variational inequality formulation of dynamic user equilibrium, equivalent to the original Friesz et al. (1993) formulation.

3.2.1 Definition of DUE with Fixed Demand In the case of fixed travel demand, an origin-destination demand matrix is given. The demand satisfaction constraint is expressed using the following flow conservation constraints:   tf . hp (t) dt = Qij ∀(i, j ) ∈ W (3.8) p∈Pij

t0

where (3.27) consists of Lebesgue integrals. Using the notation and concepts we have mentioned, the feasible region for DUE when effective delay operators are

96

3 The Variational Inequality Formulation of Dynamic User Equilibria

known is ⎧ ⎨   tf .Λ = h≥0: hp (t) dt = Qij ⎩ t0

∀ (i, j ) ∈ W

p∈Pij

⎫ ⎬ ⎭

  |P | ⊆ L2+ t0 , tf (3.9)

The following definition of dynamic user equilibrium was first articulated by Friesz et al. (1993): Definition 3.1 (Dynamic User Equilibrium with Fixed Demand) A vector of departure rates (path flows) .h∗ ∈ Λ is a dynamic user equilibrium with fixed demand if h∗p (t) > 0, p ∈ Pij ⇒ Ψp (t, h∗ ) = vij (h∗ )

∀ν (t) ∈ [t0 , tf ]

.

(3.10)

∗ where the minimum effective  delay .vij (h ) is defined by (3.6) and (3.7). We denote this equilibrium by .DU E Ψ, Λ, [t0 , tf ] .

As a contrapositive of (3.10), it is immediately clear that if .Ψp (t, h∗ ) > vij (h∗ ) for some t and p, then .h∗p (t) = 0 must hold. For notational convenience, we will write .vp∗ = vp (h∗ ) and .vij∗ = vij (h∗ ) in subsequent discussions.

3.2.2 Variational Inequality Problem Equivalent to DUE with Fixed Demand In order to establish an equivalence between a variational inequality problem and .DU E Ψ, Λ, [t0 , tf ] , we first record the following elementary property of measurable functions   Lemma 3.1 For any set .S ⊆ t0 , tf with .ν(S) > 0 and any measurable function with .ν{t ∈ S : f (t) > 0} > 0, there is some .0 > 0 such that .ν{t ∈ S : f (t) > } .> 0 for all . ∈ [0, 0 ]. Proof If for each .n > 0 we set Sn = {t ∈ S : 1/n < f (t)}

.

then by definition {t ∈ S : f (t) > 0} =



.

n

Sn

3.2 The VI Formulation of DUE with Fixed Demand

97

However, by the countable subadditivity of measures (Halmos, 1974, Theorem .8.C) 0 < ν{t ∈ S : f (t) > 0} = ν

 

.

 Sn

n

≤



ν(Sn ),

n

which in turn implies that .ν(Sn ) > 0 for some .n > 0. Hence, by letting .0 = 1/n > 0, we may conclude that Sn ⊆ {t ∈ S : f (t) > } ⇒ ν{t ∈ S : f (t) > 0} > 0

.

for all . ∈ [0, 0 ].



We are now ready to establish the following equivalent formulation of simultaneous route-and-departure-time (SRDT) dynamic user equilibrium (DUE). Theorem 3.1 (Infinite-Dimensional VI Equivalent to the SRDT DUE) The simultaneous route-and-departure-time choice dynamic user equilibrium expressed in Definition 3.1 is equivalent to the following variational inequality problem in the  |P | : Hilbert space . L2 [t0 , tf ] ⎫ ⎬

find h∗ ∈ Λ such that .

Ψ (h∗ ),

h − h∗ 

≥ 0

∀h ∈ Λ

⎭

  V I Ψ, Λ, [t0 , tf ]

(3.11)

where the inner product is defined as   .  Ψ (h ), h − h =

 .

∗

tf

∗

p∈P

t0

Ψp (t, h∗ )[hp (t) − h∗p (t)] dν(t)

Proof We follow the proof in Friesz et al. (1993). That proof consists of two parts. [Necessity] If .h∗ ∈ Λ is a dynamic user equilibrium, then to establish that .h∗ is a solution to the (3.11), it suffices to show that for all .h ∈  and .(i, j ) ∈ W  

tf

.

p∈Pij

t0

Ψp (t, h∗ )[hp (t) − h∗p (t)]dν(t) ≥ 0

However, because (3.9) implies that  

tf

.

p∈Pij

t0

[hp (t) − h∗p (t)]dν(t) = 0

∀(i, j ) ∈ W

(3.12)

98

3 The Variational Inequality Formulation of Dynamic User Equilibria

it follows that (3.12) is equivalent to the condition that  

tf

.

t0

p∈Pij

{Ψp (t, h∗ ) − vij∗ }[hp (t) − h∗p (t)]dν(t) ≥ 0 (i, j ) ∈ W

(3.13)

Hence, it suffices to show that for all .h ∈ , .p ∈ Pij and .(i, j ) ∈ W {Ψp (t, h∗ ) − vij∗ }[hp (t) − h∗p (t)] ≥ 0

.

∀ν (t)

(3.14)

  To do so, observe first that if (3.14) fails for any .t ∈ t0 , tf , then either .Ψp (t, h∗ ) − vij∗ < 0 or .hp (t) − h∗p (t) < 0. But by (3.10), for .ν-almost all t, .Ψp (t, h∗ ) − vp∗ ≥ 0. Moreover, by (3.10) it follows that for .ν-almost all t hp (t) < h∗p (t) ⇒ h∗p (t) > 0 ⇒ Ψp (t, h∗ ) = vij∗

.

⇒ {Ψp (t, h∗ ) − vij∗ }[hp (t) − h∗p (t)] = 0

(3.15)

Hence (3.14) follows. [Sufficiency] Next suppose that .h∗ ∈  satisfies (3.11) for all .h ∈ , and let |W | the individual components of the vector .v ∗ = (vij∗ : (i, j ) ∈ W) ∈ R+ be ∗ ∗ defined by .vij = vij (h ) for all .(i, j ) ∈ W. Our objective is to show that the pair .(h∗ , v ∗ ) yields a dynamic user equilibrium. To do so, observe first from the definitions of .vp∗ and .vij∗ above that, for all .p ∈ Pij and .ν-almost all t, we have ∗ ∗ ∗ ∗ ∗ ∗ .Ψp (t, h ) ≥ vp (h ) ≥ vij (h ) = v . Hence, (.h , v ) satisfies condition (3.10) by ij construction, and it remains to establish condition (3.10). To do so, suppose to the contrary that (3.10) fails for some .p ∈ Pij , .(i, j ) ∈ W. Then, by definition, the set Sp = {t ∈ [t0 , tf ] : h∗p (t) > 0, Ψp (t, h∗ ) − vij∗ > 0}

.

(3.16)

must have positive measure. In particular, this implies from Lemma 3.1 that for some sufficiently small value of . > 0 the subset Sp () = {t ∈ Sp : Ψp (t, h∗ ) − vij∗ > 2}

.

(3.17)

has positive measure. Since .Sp () ⊆ Sp ⇒ h∗p (t) > 0, .∀ν [t ∈ Sp ()], a second application of Lemma 3.1 shows that there exists some sufficiently small value of .δ > 0 such that Sp (, δ) = {t ∈ Sp () : h∗p (t) > δ}

.

(3.18)

has positive measure. Next, choosing any path .q ∈ Pij with .vq (h∗ ) = vij (h∗ ) (possibly with .q = p), it follows from the definition of .vp (h∗ ) that the set Tq () = {t ∈ [t0 , tf ] : Ψq (t, h∗ ) < vij∗ + }

.

(3.19)

3.2 The VI Formulation of DUE with Fixed Demand

99

also has positive measure. Finally, letting .α0 = min{ν[Sp (, δ)], .ν[Tq ()]} > 0 and observing that the Lebesgue measure is nonatomic, it follows (Halmos, 1974, Proposition 41.2) that for any choice of .α ∈ (0, α0 ), there exist subsets .Sp (, δ, α) ⊆ Sp (, δ) and .Tq (, α) ⊆ Tq () with .ν[Sp (, δ, α)] = α = ν[Tq (, α)]. Given these two sets, we now construct a vector of densities .h = (hr : r ∈ P) ∈ , which violates condition (3.11) for .h∗ . To do so, let .hr = h∗r for all .r ∈ P − {p, q}, and let .hp and .hq be defined, respectively, by  h∗p (t) − δ t ∈ Sp (, δ, α) .hp (t) = (3.20) t ∈ [t0 , tf ] \ Sp (, δ, α) h∗p  hq (t) =

.

h∗q + δ h∗q (t)

t ∈ Tq (, α) t ∈ [t0 , tf ] \ Tq (, α)

(3.21)

Note that if .p = q, then these two conditions still yield a well-defined function, .hp . To see this, observe from (3.17) that .Sp (, δ, α) ⊆ Sp () implies .Ψp (t, h∗ ) > vij∗ + 2, .∀ν [t ∈ Sp (, δ, α)], and similarly from (3.19) that .Tp (, α) ⊆ Tp () implies .Ψp (t, h∗ ) < vij∗ + , .∀ν [t ∈ Tp (, α)]. Hence, if .p = q, then we must have .ν[Sp (, δ, α) ∩ Tp (, α)] = 0, and it follows that .hp is well defined up to a set of measure zero. Thus, without loss of generality, we may henceforth assume that .p = q. With this convention, we next show that .h ∈ . To do so, observe first that for .ν-almost all .t ∈ Sp (, δ, α), we have .h∗p (t) ≥ δ ⇒ hp (t) ≥ 0. Similarly, for ∗ .ν-almost all .t ∈ Tq (, α), .hq (t) ≥ 0 ⇒ hq (t) ≥ 0. Moreover, ν[Sp (, δ, α)] = α = ν[Tq (, α)]

.

implies that  

tf

.

r∈Pij

t0





hr (t)dν(t) =

tf

hr (t)dν(t)

r∈Pij \{p,q} t0



+

tf

hp (t)dν(t) +

t0





=



tf

t0

 +

tf

t0

tf

hq (t)dν(t) t0

tf

r∈Pij \{p,q} t0

+



h∗r (t)dν(t)

h∗p (t)dν(t) − δ · α h∗q (t)dν(t) + δ · α





100

3 The Variational Inequality Formulation of Dynamic User Equilibria

=

  r∈Pij

tf

t0

h∗r (t)dν(t)

= Qij

(3.22)

Therefore, we must have .h ∈ . However, (3.20) and (3.21) also imply that    tf . Ψp (t, h∗ )[hp (t) − h∗p (t)]dν(t) = Ψp (t, h∗ )[hp (t) − h∗p (t)] t0

p∈P

Sp (,δ,α)

 + Tq (,α)

Cq (t, h∗ )[hq (t) − h∗q (t)] (3.23)

Furthermore, the construction of .Sp (, δ, α) and .Tq (, α) imply, respectively, that 

∗



∗

Ψp (t, h∗ )[−δ]dν(t)

Ψp (t, h )[hp (t) − h (t)] =

.

Sp (,δ,α)

Sp (,δ,α)





≤ − Sp (,δ,α)

 (vij∗ + 2)δ dν(t)

= − (vij∗ + 2)δα

(3.24)

and  .

Tq (,α)

Cq (t, h∗ )[hq (t) − h∗q (t)] =

 

Cq (t, h∗ )[δ]dν(t) Tq (,α)

≤ Tq (,α)



 (vij∗ + )δ dν(t)

= (vij∗ + )δα

(3.25)

Finally, by combining (3.23), (3.24), and (3.25), we may conclude that 

tf

.

p∈P

t0

Ψp (t, h∗ )[hp − h∗p (t)]dν(t) ≤ − δα < 0,

(3.26)

which contradicts (3.11) for this choice of .h ∈ . Thus the hypothesized failure of condition (3.10) leads to a contradiction, and we may conclude that the pair .(h∗ , v ∗ ) yields a simultaneous route-and-departure-time equilibrium.  In Sect. 9.1 from Chap. 9, to clarify and reinforce the notion of equivalence addressed in Theorem 3.1 and its proof, we present a DUE solution of an illustrative, simple network.

3.3 The VI Formulation of DUE with Elastic Demand

101

3.3 The VI Formulation of DUE with Elastic Demand Most of the studies of DUE reported in the dynamic traffic assignment (DTA) literature are about dynamic user equilibrium with constant travel demand for each origin-destination pair. It is, of course, not generally true that travel demand is fixed, even for short time horizons. Arnott et al. (1993) and Yang and Huang (1997) are among the earliest to model elastic demand; however, their work was conducted in the context of a single bottleneck and not directly applicable to a general network. Yang and Meng (1998) extend a simple bottleneck model to a general queuing network with known elastic demand functions for each origin-destination (OD) pair. Wie et al. (2002) study a version of dynamic user equilibrium with elastic demand, using a complementarity formulation that requires path delays to be expressible in closed form. Szeto and Lo (2004) study dynamic user equilibrium with elastic travel demand when network loading is based on the cell transmission model (CTM); their formulation is based on discrete time and is expressed as a finite-dimensional variational inequality (VI). Han et al. (2011) study dynamic user equilibrium with elastic travel demand for a network whose traffic flows are also described by CTM. Friesz and Meimand (2014) are the first to extend the DVI formalism to an elastic demand setting. In subsequent work, Han et al. (2015a) further study existence of an elastic demand DUE and reformulate it as a fixed-point problem in an extended Hilbert space. The authors also presented three different computational algorithms for the elastic demand DUE with convergence results that address non-monotone delay operators. The DVI formulation of elastic demand DUE is not straightforward. In particular, an elastic demand DVI has both infinite-dimensional and finite-dimensional terms; moreover, for any given origin-destination pair, inverse travel demand corresponding to a dynamic user equilibrium requires a dual variable to be found. The DVI formulation achieved in this book is significant because it allows the still emerging theory of differential variational inequalities to be employed for the analysis and computation of solutions of the elastic-demand DUE problem when simultaneous departure time and route choice are within the purview of users, all of which constitutes a foundation problem within the field of dynamic traffic assignment. A good review of recent insights into abstract differential variational inequality theory, including computational methods for solving such problems, is provided by Pang and Stewart (2008). Also, differential variational inequalities involving the kind of explicit, agent-specific control variables employed herein are presented in Friesz (2010).

3.3.1 Definition of DUE with Elastic Demand   We introduce the trip matrix . Qij : (i, j ) ∈ W , where now each .Qij ≥ 0 is the elastic travel demand between the origin-destination (O-D) pair .(i, j ) ∈ W, and

102

3 The Variational Inequality Formulation of Dynamic User Equilibria

W is the set of all origin-destination (O-D) pairs. The flow conservation constraints read   tf . hp (t) dt = Qij ∀(i, j ) ∈ W (3.27)

.

p∈Pij

t0

where (3.27) consists of Lebesgue integrals, and .Pij ⊂ P is the set of paths connecting O-D pair .(i, j ) ∈ W. For each O-D pair .(i, j ) ∈ W, the travel demand is assumed to be expressed as the following invertible function Qij = Gij [v]

(3.28)

.

where .v = {νij (h) : (i, j ) ∈ W} is a vector of O-D minimum travel costs .νij , which depend on the path flow vector h. Note that to say .vij is a minimum travel cost means it is the minimum cost for all departure time choices and all route choices pertinent to origin-destination pair .(i, j ) ∈ W; see (3.6) and (3.7). Further note that .Qij is the unknown cumulative travel demand between .(i, j ) ∈ W that must ultimately arrive by time .tf . We will also find it convenient to form the complete vector of travel demands by concatenating the OD-specific travel demands to obtain Q =

.

  |W | Qij : (i, j ) ∈ W ∈ R+

The inverse demand function for every .(i, j ) ∈ W is vij = Θij [Q]

(3.29)

.

and we naturally define Θ[Q] =

.

  |W | |W | Θij [Q] : (i, j ) ∈ W : R+ → R++

We employ the following feasible set of departure flows when the travel demand between each origin-destination pair is unknown. = Λ

.

⎧ ⎨ ⎩

(h, Q) : h ≥ 0,

  p∈Pij

 |P | ⊂ L2 [t0 , tf ] × R|W |

tf

t0

hp (t) dt = Qij

∀(i, j ) ∈ W

⎫ ⎬ ⎭ (3.30)

 |P | where . L2 [t0 , tf ] × R|W | is the direct product of the .|P|-fold product of Hilbert spaces consisting of square-integrable path flows, and the .|W|-dimensional Euclidean space consisting of vectors of elastic travel demands. With the preceding

3.3 The VI Formulation of DUE with Elastic Demand

103

preparation, we are now ready to formally define the simultaneous route-anddeparture-time choice dynamic user equilibrium with elastic demand: Definition 3.2 (Dynamic User Equilibrium with Elastic Demand) A pair  where .h∗ is a vector of departure rates (path flows) and .Q∗ is (h∗ , Q∗ ) ∈ Λ, the associated vector of travel demands, is said to be a dynamic user equilibrium with elastic demand if for all .(i, j ) ∈ W,

.

h∗p (t) > 0, p ∈ Pij ⇒ Ψp (t, h∗ ) = Θij [Q∗ ]

∀ν (t) ∈ [t0 , tf ]. (3.31)

.

Ψp (t, h∗ ) ≥ Θij [Q∗ ]

∀ν (t) ∈ [t0 , tf ] ∀p ∈ Pij

(3.32)

According to Definition 3.2, a vector of path departure rates .h∗ yields an elastic demand DUE (E-DUE) if (1) it is a normal DUE with simultaneous route-anddeparture-time choices; and (2) the minimum travel cost .vij (h∗ ) for each O-D pair .(i, j ) is related to the travel demands   .Q = Q∗ij : (i, j ) ∈ W , ∗

Q∗ij

. =

  p∈Pij

tf t0

h∗p (t) dt

in a way consistent with the pre-defined inverse demand function .Θij [·]. An analytical and closed-form example of the E-DUE will be provided later in Sect. 9.1.3.

3.3.2 Variational Inequality Problem Equivalent to DUE with Elastic Demand Experience with differential games suggests that the DUE problem with elastic demand can be expressed as a variational inequality, as shown in the theorem below. Theorem 3.2 (DUE with Elastic Demand Equivalent to a Variational Inequality Problem) We stipulate that .Ψp (·, h) : [t0 , tf ] → R++ is measurable and positive  We also stipulate almost everywhere for all .p ∈ P and all h such that .(h, Q) ∈ Λ. that the travel demand defined in (3.28) is elastic and strictly monotone, so that the inverse demand .Θij (·) exits and is well defined for all .(i, j ) ∈ W. Then a pair ∗ ∗  is a DUE with elastic demand as in Definition 3.2 if and only if it .(h , Q ) ∈ Λ solves the following variational inequality:  .

p∈P

tf

t0

⎫  such that find (h∗ , Q∗ ) ∈ Λ ⎪ ⎪ ⎪   ⎬  ∗  ∗ ∗ ∗ Ψp (t, h )(hp (t) − hp (t)) dt − Θij Q Qij − Qij ≥ 0 ⎪ ⎪ (i, j )∈W ⎪ ⎭  ∀(h, Q) ∈ Λ (3.33)

104

3 The Variational Inequality Formulation of Dynamic User Equilibria

 we Proof [Necessity]. Given a DUE solution with elastic demand .(h∗ , Q∗ ) ∈ Λ,  easily deduce from (3.31) and (3.32) that for any .(h, Q) ∈ Λ, 

tf

.

t0

p∈P

=

Ψp (t, h∗ )(hp (t) − h∗p (t))dt −

⎛



⎝

(i, j )∈W

  ⎛

=



⎝

(i, j )∈W

−

  ⎛

=

⎛



⎝

(i, j )∈W

Ψp (t, h∗ )hp (t) dt − Θij [Q∗ ] · Qij ⎠

Θij [Q∗ ] ·

 

p∈Pij

=

(i, j )∈W

⎝

Ψp (t, h∗ )hp (t) dt − Θij [Q∗ ] · Qij ⎠

t0

⎛

 

tf

⎞

 

tf

t0

p∈Pij

p∈Pij

h∗p (t) dt − Θij [Q∗ ] · Q∗ij ⎠ ⎞

tf

(i, j )∈W



tf

t0

Θij [Q∗ ] · ⎝

⎛

⎞



p∈Pij



−

⎞

tf



⎝

Ψp (t, h∗ )h∗p (t) dt − Θij [Q∗ ] · Q∗ij ⎠

t0

p∈Pij

(i, j )∈W

⎞ tf t0

p∈Pij



⎞

Ψp (t, h∗ )hp (t) dt − Θij [Q∗ ] · Qij ⎠

 

⎝

(i, j )∈W

tf t0

p∈Pij

⎛

  Θij Q∗ (Qij − Q∗ij )

(i, j )∈W



−



h∗p (t) dt − Q∗ij ⎠ ⎞

Ψp (t, h∗ )hp (t) dt − Θij [Q∗ ] · Qij ⎠

(3.34)

t0

We next observe that for any .(i, j ) ∈ W,  

tf

.

p∈Pij ∗

≥ vij (h )

Ψp (t, h∗ )hp (t) dt − Θij [Q∗ ] · Qij

t0

  p∈Pij

⎛ = vij (h∗ ) ⎝

tf

  p∈Pij

hp (t) dt − Θij [Q∗ ] · Qij

t0

⎞ tf

t0

hp (t) dt − Qij ⎠ = 0

(3.35)

3.3 The VI Formulation of DUE with Elastic Demand

105

where .vij (h∗ ) is the essential infimum of the effective path delay associated with O-D .(i, j ) and is equal to .Θij [Q∗ ] according to Definition 3.2. As an immediate  consequence of (3.34) and (3.35), the following inequality holds for all .(h, Q) ∈ Λ: 

tf

.

p∈P

t0



Ψp (t, h∗ )(hp (t) − h∗p (t))dt −

Θij [Q∗ ](Qij − Q∗ij ) ≥ 0

(i, j )∈W

(3.36) which is the desired variational inequality.  Then .h∗ must be [Sufficiency]. Assume that (3.36) holds for any .(h, Q) ∈ Λ. a solution of the fixed-demand DUE problem with fixed demand given by .Q∗ . The second term on the left-hand side of (3.36) vanishes, and we recover the well-known VI for the fixed demand case; see Theorem 2 of Friesz et al. (1993). By definition of fixed-demand DUE, for any .(i, j ) ∈ W, .

h∗p (t) > 0, p ∈ Pij ⇒ Ψp (t, h∗ ) = vij (h∗ ) ∀ν (t) ∈ [t0 , tf ] ∗ ∗ ∀p ∈ Pij , ∀ν (t) ∈ [t0 , tf ] Ψp (t, h ) ≥ vij (h )

(3.37)

In order to show that .(h∗ , Q∗ ) is a DUE with elastic demand using Definition 3.2, we fix an arbitrary O-D pair .(k, l) ∈ W and distinguish between two cases: ˆ Q) ˆ ∈ Λ:  • .Q∗kl > 0. We define the following pair .(h,  ˆ p (t) = .h  ˆ ij = Q

a h∗p (t)

p ∈ Pkl

h∗p (t)

p ∈ P \ Pkl

a Q∗kl

(i, j ) = (k, l)

Q∗ij

∀t ∈ [t0 , tf ],

(i, j ) ∈ W \ (k, l)

ˆ Q), ˆ where .a > 0 is an arbitrary positive parameter. Substituting .(h, Q) for .(h, the left-hand side of (3.36) becomes 

tf

.

p∈P

=

t0

 

p∈Pkl

Ψp (t, h∗ )(hˆ p − h∗p )dt −



  ˆ ij − Q∗ij ) Θij Q∗ (Q

(i, j )∈W tf

t0

Ψp (t, h∗ )(ah∗p − h∗p ) dt +

  − Θkl [Q∗ ] aQ∗kl − Q∗kl −



 p∈P \Pkl

(i, j )∈W \(k, l)



tf

t0

Ψp (t, h∗ )(h∗p − h∗p ) dt

Θij [Q∗ ](Q∗ij − Q∗ij )

106

3 The Variational Inequality Formulation of Dynamic User Equilibria

= (a − 1)

  p∈Pkl

tf

Ψp (t, h∗ ) h∗p (t) dt − (a − 1)Θkl [Q∗ ] Q∗kl

t0

= (a − 1)vkl (h∗ )Q∗kl − (a − 1)Θkl [Q∗ ]Q∗kl We conclude from (3.36) that   (a − 1) vkl (h∗ ) − Θkl [Q∗ ] Q∗kl ≥ 0

.

Since .a > 0 is arbitrary and .Q∗kl > 0, there must hold .vkl (h∗ ) = Θkl [Q∗ ]. Thus replacing .vkl (h∗ ) with .Θkl [Q∗ ] in (3.37) yields (3.31)-(3.32). ˆ Q) ˆ ∈Λ  such that .hp (t) ≡ h∗p (t) for all • .Q∗kl = 0. In this case, we consider .(h, ˆ Q) ˆ in (3.36) yields .t ∈ [t0 , tf ], .p ∈ P \ Pkl . Substituting .(h, Q) for .(h,  

tf

Ψp (t, h∗ )(hp (t) − h∗p (t))dt − Θkl [Q∗ ]Qkl ≥ 0,

.

p∈Pkl

t0

(3.38)

which implies that  

tf

.

p∈Pkl

  Ψp (t, h∗ ) − Θkl [Q∗ ] hp (t) dt ≥ 0

t0

Since the vector .(hp (t) : p ∈ Pkl , t ∈ [t0 , tf ]) is arbitrary, we must have that Ψp (t, h∗ ) ≥ Θkl [Q∗ ] for any .p ∈ Pkl and almost every .t ∈ [t0 , tf ], which is (3.32). Finally, (3.31) is trivially true since .h∗p (t) ≡ 0 for all .p ∈ Pkl and almost every .t ∈ [t0 , tf ].

.

Combining the above two cases, we finish the proof.



Remark 3.2 The problem identified in (3.33) can be viewed as a generic variational inequality problem defined in an extended (product) Hilbert space, as we subsequently illustrate in Sect. 4.4 and Chap. 5. Such a VI form is crucial for the existence and computation of E-DUE problems. In Sect. 9.1 from Chap. 9, we will present a DUE with elastic demand on a simple network with closed-form expressions and illustrate its equivalence to the variational inequality (3.33).

3.4 The VI Formulation of Boundedly Rational DUE In dynamic traffic assignment (DTA) problems, the modeling of travelers’ routeand-departure-time choices have been greatly influenced by Wardrop’s first principle (Wardrop, 1952), originally conceived for static equilibria. Flows satisfying

3.4 The VI Formulation of Boundedly Rational DUE

107

Wardrop’s first principle are usually said to constitute a user equilibrium (UE) wherein travelers adjust their routes with complete rationality until no lower disutility can be achieved. When UE behavior is generalized to include departure time choice, the result is, as we have stated previously, one type of dynamic user equilibrium (DUE), which is based on complete rationality. While admitting a number of canonical mathematical representations, the notion of completely rational user equilibrium is not consistent with observed driver behavior. That is, travelers may not always choose the departure time and route that yield the minimum travel cost. Such a situation could be caused by (1) imperfect travel information and (2) certain “inertia” in decision-making. Moreover, empirical studies suggest that in reality, drivers do not always follow the least costly route-and-departure-time choice (Avineri and Prashker, 2004). As a relaxation of the perfect rationality assumption, the notion of bounded rationality is proposed by Simon (1957, 1990, 1991) and introduced to traffic modeling by Mahmassani and Chang (1987). In prose, the notion of bounded rationality postulates a range of acceptable travel costs that, when achieved, do not incentivize travelers to change their departure times or route choices. Such a range is described by Mahmassani and Chang (1987) as an “indifference band.” The width of such a band, usually denoted by .ε, is either derived through a behavioral study of road users (e.g., by surveys) or calibrated from empirical observation through inverse modeling techniques. In general, .ε could depend on a specific origin-destination pair and/or travel commodity. Boundedly rational user equilibrium has gradually become a well-recognized field of inquiry, especially in static traffic assignment (STA). Among the recent published works addressing boundedly rational user equilibrium (BR-UE) from an analytical, theoretical, and/or computational perspective are Di et al. (2013), Gifford and Checherita (2007), Han and Timmermans (2006), Khisty and Arslan (2005), Luo et al. (2010), and Marsden et al. (2012). Bounded rationality is also investigated via simulation-based approaches of dynamic modeling (Hu and Mahmassani, 1997; Mahmassani and Chang, 1991a,b; Mahmassani et al., 2005). The notion of bounded rationality (BR) was used somewhat imprecisely during the early days of dynamic traffic assignment research. In particular, BR was studied in a so-called laboratory setting by Mahmassani and Chang (1987), but without a precise mathematical articulation of BR for dynamic traffic assignment. BR was used in a similarly ad hoc fashion for simulations by Jayakrishnan and Mahmassani (1990), Peeta and Mahmassani (1995), Mahmassani and Chang (1991b), and Chiu and Mahmassani (2002). Recognizing the lack of a theory of traffic assignment that directly incorporates BR, Ridwan (2004) tried to apply the theory of fuzzy systems to the study of BR. Bogers et al. (2005), again driven by the lack of a suitable theory, conducted more laboratory studies of BR. Szeto (2003) and Szeto and Lo (2006) propose route-choice (RC) boundedly rational dynamic user equilibrium (RC BR-DUE). The RC BR-DUE problem is formulated as a discrete-time nonlinear complementarity problem in Szeto and Lo (2006), where a heuristic route-swapping algorithm is proposed to solve the problem. Ge and Zhou (2012) consider RC BR-DUE with endogenously determined tolerances by allowing the width of the

108

3 The Variational Inequality Formulation of Dynamic User Equilibria

indifference band .ε to depend on time and the actual path flows. However, no existence or computational method is provided in that paper. Contributions by Szeto (2003), Szeto and Lo (2006), and Ge and Zhou (2012) achieved enhanced (yet partial) integration of BR and DUE but did not establish and analyze a complete theory, where by “complete” we mean a mathematical formulation consistent with known empirical results, surmised behaviors, qualitative properties, and a computational approach that is demonstrably effective.

3.4.1 Definition of DUE with Bounded Rationality The notion of bounded rationality (BR) is a relaxation of the dynamic extension of Wardrop’s first principle (Wardrop, 1952), which stipulates equal travel costs among all utilized route-and-departure-time choices between an origin-destination pair. The bounded rationality dynamic user equilibrium (BR-DUE), on the other hand, requires that the experienced travel cost, including early and late arrival penalties, are within the interval .[vij (h), vij (h) + εij ]. Here .vij (h) is the minimum effective delay (travel cost) for origin-destination  pair .(i, j ), and  it depends on the complete vector of path departure rates .h = hp (·) : p ∈ P . Furthermore, .εij ∈ R+ is a prescribed constant describing acceptable differences in travel costs experienced by travelers between origin-destination pair .(i, j ). Notice that, in most literature on BR-DUE, .εij is an exogenously given constant and is usually estimated from static attributes such as the socio-economic characteristics of travelers between a given O-D pair. As an extension, we will also investigate a more general case where the tolerances depend not only on the O-D pair, but also on the path p and the established path departure rates h. In this case, the tolerance is endogenous to the traffic system as it depends on the unknown variable h. We refer to such a generalization of boundedly rational dynamic user equilibrium as a variable tolerance BR-DUE, or VT-BR-DUE. Throughout this section, the travel demand .Qij between each origin-destination pair .(i, j ) is assumed to be a fixed constant, so that we are dealing with fixeddemand problems. We begin by articulating the notion of bounded rationality in a measure-theoretic context. Recall that the effective delay operator .Ψ maps each vector of path departure rates .h ∈ Λ to a vector of effective path delays, that is,  |P |  |P | Ψ : L2+ [t0 , tf ] → L2+ [t0 , tf ] .

h(·) = {hp (·), p ∈ P} → Ψ (h) = {Ψp (·, h), p ∈ P}

3.4 The VI Formulation of Boundedly Rational DUE

109

Recall that the set of feasible path flows is: ⎧ ⎨

Λ =

.

⎩

h≥0:

  p∈Pij

tf

hp (t) dt = Qij

t0

∀ (i, j ) ∈ W

⎫ ⎬ ⎭

 |P | ⊆ L2+ [t0 , tf ] (3.39)

The following is a precise definition of boundedly rational dynamic user equilibrium (BR-DUE) with exogenous tolerances and simultaneous route-and-departure-time choices.  Definition 3.3 (BR-DUE) Given the vector of fixed tolerances .ε = εij : (i, j ) ∈  |W | W ∈ R+ , a vector of departure rates .h∗ ∈ Λ is a BR-DUE associated with .ε if, for all .(i, j ) ∈ W, h∗p (t) > 0, p ∈ Pij ⇒ Ψp (t, h∗ ) ∈ [vij (h∗ ), vij (h∗ ) + εij ]

.

∀ν (t) ∈ [t0 , tf ] (3.40)

where .vij (h∗ ) is the essential infimum of the effective path delays between origindestination  pair .(i, j ) and is defined in (3.6)–(3.7). We denote this equilibrium by BR-DUE. Ψ, ε, Λ, t0 , tf .   In Definition 3.3 the tolerance vector . εij : (i, j ) ∈ W is exogenously given. In the following definition, the tolerances can vary according to the path and the p established path flows. In particular, we introduce .εij (h) ∈ R+ for .(i, j ) ∈ W,  .  p .p ∈ Pij , and .h ∈ Λ. We define .ε(h) = ε (h) : (i, j ) ∈ W, p ∈ Pij to be the ij concatenation of tolerances associated with each path and O-D pair. Note that .ε(·) |P | is viewed as a mapping from .Λ to .R+ . |P |

Definition 3.4 (VT-BR-DUE) Given the vector of tolerances .ε(·) : Λ → R+ , which is viewed as a mapping, a vector of departure rates .h∗ ∈ Λ is a VT-BR-DUE associated with .ε(h) if, for all .(i, j ) ∈ W, h∗p (t) > 0, p ∈ Pij ⇒ Ψp (t, h∗ ) ∈ [vij (h∗ ), vij (h∗ ) + εij (h∗ )] p

.

∀ν (t) ∈ [t0 , tf ]

(3.41)

where .vij (h∗ ) is the essential infimum of the effective path delays associated with origin-destination pair .(i, j) and is defined by (3.6) and (3.7). We denote this equilibrium by VT-BR-DUE. Ψ, ε, Λ, t0 , tf . Remark 3.3 It can be easily seen that the BR-DUE is just one special case of the VTp BR-DUE problem, in which the dependence of .εij (h) on p and h are dropped. Thus, to simultaneously analyze both models using the proposed framework, it suffices for us to treat the VT-BR-DUE only, and the established results will automatically hold for the BR-DUE problem.

110

3 The Variational Inequality Formulation of Dynamic User Equilibria

3.4.2 Variational Inequality Equivalent to BR-DUE with Exogenous or Endogenous Tolerances In this section, we present the infinite-dimensional variational inequality (VI) formulations for the BR-DUE and VT-BR-DUE problems, when both route choice and departure time choice are considered. These formulations, first presented in Han et al. (2015b), will be central to the study of existence and computation of VT-BRDUE given subsequently in this book. For the reason stated in Remark 3.3, we will first present and show the validity of a variational inequality formulation of VT-BRDUE and then present a corollary that establishes the same for BR-DUE. Essential to the variational inequality formulation of the VT-BR-DUE problem is the following operator not previously considered:  |P | Φ ε : Λ → L2+ [t0 , tf ] ,

.

  h → Φpε (·, h) : p ∈ P

(3.42)

where " " # $ #% p p q Φpε (t, h) = max Ψp (t, h), vij (h) + εij (h) − εij (h) − min εij (h)

.

q∈Pij

∀p ∈ Pij

(3.43)

Given any vector of path departure rates .h ∈ Λ, by performing the dynamic network loading procedure, one obtains the effective path delays .Ψp (t, h), ∀t ∈ [t0 , tf ], ∀p ∈ P. Thus the essential infimum .vij (h) can be determined for each p O-D pair .(i, j ). Moreover, the path-specific variable tolerances .εij (h) may also be determined from h. Consequently, one can readily construct the quantities ε ε .Φp (t, h), ∀t ∈ [t0 , tf ], ∀p ∈ P according to (3.43). Thus, the operator .Φ stated  2 |P | . We indicate above can indeed be viewed as a mapping from .Λ into . L+ [t0 , tf ] the dependence of such an operator on the variable tolerances .ε(·) by a superscript. Theorem 3.3 below casts the VT-BR-DUE problem as an infinite-dimensional variational inequality. In comparison with the VI formulation of DUE presented in Theorem 3.1, this new variational inequality relies on the new principal operator .Φ ε , which encapsulates both the DNL procedure and the variable tolerances. Notably, such a formulation allows known methodologies regarding variational inequalities to be directly applied to VT-BR-DUE problems. Theorem 3.3 (VT-BR-DUE Equivalent to a Variational Inequality) Given .ε(·) : |P | Λ → R+ , define .Φ ε (·) according to (3.42)-(3.43). Then, a vector of path departure rates .h∗ ∈ Λ is a VT-BR-DUE solution if and only if it solves the following

3.4 The VI Formulation of Boundedly Rational DUE

111

variational inequality.  .

p∈P

tf

t0

find h∗ ∈ Λ such that Φpε (t, h∗ )(hp (t) − h∗p (t)) dt ≥ 0 ∀h∈Λ

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

  V I Φ ε , Λ, [t0 , tf ]

(3.44)

Proof [Necessity] Let .h∗ ∈ Λ be a VT-BR-DUE solution. Then for any .h ∈ Λ and any .(i, j ) ∈ W, we have  



tf

(i, j )∈W p∈Pij



=

t0

μεij (h∗ )

(i, j )∈W



Φpε (t, h∗ ) hp (t) dt ≥

.

  p∈Pij

 

(i, j )∈W p∈Pij tf

hp (t) dt =

t0



tf

t0

μεij (h∗ ) hp (t) dt

μεij (h∗ )Qij

(3.45)

(i, j )∈W

where " # . μεij (h∗ ) = min μεp (h∗ ) ,

.

p∈Pij

" # . μεp (h∗ ) = essinf Φpε (t, h∗ ) t∈[t0 , tf ]

(3.46)

We claim that " # q μεij (h∗ ) = vij (h∗ ) + min εij (h∗ )

.

(3.47)

q∈Pij

Indeed, we first notice from (3.43) that μεp (h∗ )

.

= essinf

t∈[t0 , tf ]

& " #'   p p q max Ψp (t, h∗ ), vij (h∗ ) + εij (h∗ ) − εij (h∗ ) + min εij (h∗ )

& = essinf

t∈[t0 , tf ]

 p max Ψp (t, h∗ ), vij (h∗ ) + εij (h∗ )

' 

q∈Pij

" # p q − εij (h∗ ) + min εij (h∗ ) q∈Pij

  We then distinguish two cases. If .p ∈ Pij is such that . essinf Ψp (t, h∗ ) ≥ p vij (h∗ ) + εij (h∗ ),

t∈[t0 , tf ]

then

μεp (h∗ )

.

" " # #   p q q = essinf Ψp (t, h∗ ) − εij (h∗ ) + min εij (h∗ ) ≥ vij (h∗ ) + min εij (h∗ ) t∈[t0 , tf ]

q∈Pij

q∈Pij

(3.48)

112

3 The Variational Inequality Formulation of Dynamic User Equilibria

  p On the other hand, if .p ∈ Pij is such that . essinf Ψp (t, h∗ ) < vij (h∗ ) + εij (h∗ ), t∈[t0 , tf ]

then μεp (h∗ )

.

" " # # p p q q = vij (h∗ ) + εij (h∗ ) − εij (h∗ ) + min εij (h∗ ) = vij (h∗ ) + min εij (h∗ ) q∈Pij

q∈Pij

(3.49) Expressions (3.48) and (3.49) in combination require " " # # q μεij = min μεp (h∗ ) ≥ vij (h∗ ) + min εij (h∗ )

.

p∈Pij

q∈Pij

Finally, notice that there exists at least one path p such that (3.49) is true; one such ∗ path is that for which the essential " #infimum .vij (h ) is attained. We thus conclude that .μεij = vij (h∗ ) + min εij (h∗ ) . Our claim (3.47) is substantiated. q

q∈Pij

In view of (3.41) and (3.43), we perform the following deduction for every .p ∈ Pij : h∗p (t) > 0 ⇒ Ψp (t, h∗ ) ≤ vij (h∗ ) + εij (h∗ ) p

.

⇒

Φpε (t,

∗

h ) = vij (h

∗

p ) + εij (h) −

$ " #% p ∗ q ∗ εij (h ) − min εij (h ) q∈Pij

= μεij (h∗ ) Therefore, according to the non-negativity of h, we have 

 

tf

.

t0

(i, j )∈W p∈Pij

Φpε (t,

h

∗

) h∗p (t) dt

=

 



t0

(i, j )∈W p∈Pij

=



tf

μεij (h∗ ) h∗p (t) dt

μεij (h∗ )Qij

(3.50)

(i, j )∈W

In view of (3.45) and (3.50), we have 

 

tf

.

(i, j )∈W p∈Pij

t0

Φpε (t, h∗ ) h∗p (t) dt ≤



 

(i, j )∈W p∈Pij

∀h ∈ Λ, which is recognized as the variational inequality (3.44).

tf

t0

Φpε (t, h∗ ) hp (t) dt

3.4 The VI Formulation of Boundedly Rational DUE

113

[Sufficiency] Let .h∗ ∈ Λ be a solution of the variational inequality. Clearly, h) is measurable and positive for any .p ∈ P and any .h ∈ Λ. We invoke the same proof of Theorem 2 from Friesz et al. (1993) to show that .h∗ satisfies

ε .Φp (·,

h∗p (t) > 0, p ∈ Pij ⇒ Φpε (t, h∗ ) = μεij (h∗ )

∀ν (t) ∈ [t0 , tf ],

.

∀(i, j ) ∈ W (3.51)

where .μεij (h∗ ) is given by (3.46) and is equal to .vij (h∗ ) + minq∈Pij {εij (h∗ )}. We readily deduce that q

h∗p (t) > 0, p ∈ Pij

.

⇒ Φpε (t, h∗ ) = vij (h∗ ) + min {εij (h∗ )} p

p∈Pij

" # p p q

⇒ max Ψp (t, h∗ ), vij (h∗ ) + εij (h∗ ) − εij (h∗ ) + min {εij (h∗ )} q∈Pij

∗

= vij (h ) + min "

q∈Pij

q {εij (h∗ )}

⇒ max Ψp (t, h∗ ), vij (h∗ ) + εij (h∗ ) p

#

= vij (h∗ ) + εij (h∗ ) p

⇒ vij (h∗ ) ≤ Ψp (t, h∗ ) ≤ vij (h∗ ) + εij (h∗ ) p

for almost every .t ∈ [t0 , tf ], .∀p ∈ Pij , .∀(i, j ) ∈ W. Therefore, .h∗ solves the VT-BR-DUE problem.  As a special case of Theorem 3.3, we now present the VI formulation for the BR  |W | DUE problem with exogenously given tolerances .ε = εij : (i, j ) ∈ W ∈ R+ . Corollary 3.1 (BR-DUE Equivalent to a Variational Inequality) Given the fixed   |W | tolerance vector .ε = εij : (i, j ) ∈ W ∈ R+ , define   . φpε (t, h) = max Ψp (t, h), vij (h) + εij

∀p ∈ Pij ,

.

∀h ∈ Λ

(3.52)

Then, a path departure rate vector .h∗ ∈ Λ is a BR-DUE solution if and only if it solves the following variational inequality  .

p∈P

tf

t0

find h∗ ∈ Λ such that φpε (t, h∗ )(hp (t) − h∗p (t)) dt ≥ 0 ∀h∈Λ

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

  V I φ ε , Λ, [t0 , tf ]

(3.53)

Proof In the case with fixed tolerances .εij for all .(i, j ) ∈ W, the operator .Φpε defined in (3.43) reduces to (3.52), and (3.44) yields (3.53). 

114

3 The Variational Inequality Formulation of Dynamic User Equilibria

We provide below an intuitive and graphical illustration of the VI formulations for DUE and BR-DUE. For simplicity, we consider just one O-D pair with one path p. Figure 3.1 depicts a DUE solution .h∗p (·) and the corresponding effective path delay .Ψp (·, h∗p ), both as functions of departure time t. The VI formulation for DUE is equivalent to 

tf

.

Ψp (t, h∗ )hp (t) dt ≥

t0



tf t0

Ψp (t, h∗ )h∗p (t) dt

∀h ∈ Λ

In other words, .h∗ is the minimizer of 

tf

.

Ψp (t, h∗ ) hp (t) dt

(3.54)

t0

(t among all .hp (·) that satisfies . t0f hp (t) dt = Qij . Therefore, the time t such that ∗ .hp (t) > 0 should be along the “flat bottom” of the effective delay curve; see Fig. 3.1. In order to minimize the quantity (3.54) over all .hp (·) ∈ Λ, the equilibrium solution ∗ .hp (·) must be located at the “flat bottom” of the effective delay curve. This coincides with the definition of DUE. To satisfy the BR-DUE condition, .h∗p (·) must reside within the time interval ε ∗ .[a, b] (left). On the other hand, the function .φp (t, h ), defined in (3.52), is represented by the thick black curve in the figure on the right. Thus, a departure rate ∗ .hp (·) satisfies the BR-DUE condition if and only if it is located at the “flat bottom” of the curve .φpε (·, h∗ ). On the other hand, the BR-DUE condition requires that whenever the solution .h∗p (t) is non-zero, .Ψp (t, h∗ ) must be within an indifference band .[vij (h∗ ), vij (h∗ ) + εij ]. This means that the positive portion of .h∗p (·) must

h*p (t)

Ψp (t, h* )

t Fig. 3.1 An illustration of a DUE solution .h∗p (·) and the associated effective path delay .Ψp (·, h∗ )

3.5 Kuhn-Tucker Conditions for DUE Problems Ψp (t, h* )

h *p (t)

115 ε

φ p (t, h* )

h *p (t)

ν ij (h* ) + ε

ν ij (h* ) + ε

ν ij (h* )

ν ij (h* )

ij

ij

a

b

t

a

b

t

Fig. 3.2 An illustration of a BR-DUE solution .h∗p (·) and the associated effective path delay ∗ .Ψp (·, h ) (left)

reside within the time interval .[a, b] (as in Fig. 3.2, left). This immediately implies that the graph of .h∗p (·) must reside within the “flat bottom” of the function .φpε (·, h∗ ) defined in (3.52), as in Fig. 3.2, right. This leads to a new VI whose principal operator is .φpε (·, h∗ ).

3.5 Kuhn-Tucker Conditions for DUE Problems The so-called Kuhn-Tucker necessary conditions for finite-dimensional variational inequalities, as we presented in Sect. 2.2.5 of Chap. 2, express the gradient as a linear combination of binding constraints at optimality. As we shall explore in this section, the Kuhn-Tucker type necessary conditions may also be applied to DUE models.

3.5.1 Application to DUE with Fixed Demand We begin by presenting a discrete-time DUE expressed as a finite-dimensional variational inequality problem. In the subsequent discussion, we will employ appropriate discrete-time approximations of integrals or derivatives whenever necessary.

3.5.1.1

Kuhn-Tucker Conditions for Discrete-time DUE Problems with Fixed Demand

Considering the same planning horizon .[t0 , tf ], we introduce a uniform time grid with step size .Δt: t0 = t1 < . . . < tk−1 < tk < tk+1 < tn = tf

.

116

3 The Variational Inequality Formulation of Dynamic User Equilibria

The flow conservation constraints are expressed using such a time grid as  .

Δt

p∈Pw

n 

hkp = Qw

∀w ∈ W

(3.55)

k=1

where .hkp is interpreted as the discrete value of path flow .hp (·) at .tk . For reasons that will become clear later, it is convenient to use w instead of .(i, j ) to denote an origin-destination pair. In the above identity, we employ rectangular quadrature for n×|P | approximating integrals. Let us also introduce the vector .h ∈ R+ that is viewed as the concatenation of .|P| vectors .hp1 , . . . , hp|P| where each .hpi ∈ Rn+ . Then the set of feasible path flows is expressed as . Λd =

.

⎧ ⎨ ⎩

h ∈ Rn×|P | : h ≥ 0,



Δt

p∈Pw

n 

hkp = Qw

∀w ∈ W

k=1

⎫ ⎬ ⎭ (3.56)

where the subscript d is used to indicate the discrete-time nature of our problem. Define the mapping .Ψ among finite-dimensional vector spaces: n×|P |

Ψ : R++

.

n×|P |

→ R++ ,

  h → Ψpk : p ∈ P, k = 1, . . . , n

(3.57)

where .Ψpk denotes the effective delay of drivers who depart at time .tk and travel along path p. The following theorem is straightforward. Theorem 3.4 (Discrete-Time DUE with Fixed Demand Equivalent to a FiniteDimensional VI) The simultaneous route-and-departure-time dynamic user equilibrium with fixed demand expressed in discrete time is equivalent to the following finite-dimensional variational inequality problem

.

find h∗ ∈ Λd such that

⎫ ⎪ ⎬

 T Ψ (h∗ ) (h − h∗ ) ≥ 0

⎪ ∀h ∈ Λd ⎭

  V Id Ψ, Λd , t0 , tf

(3.58)

It is important to realize that the feasible set .Λd is defined in terms of linear equality constraints (3.55) and convex inequality constraints .−h ≤ 0. Therefore, the Kuhn-Tucker conditions are both necessary and sufficient for optimality. The following theorem is obtained as an immediate consequence of Theorems 2.6 and 2.7 from Chap. 2. Theorem 3.5 (Sufficient and Necessary KT Conditions for DUE with Fixed Demand) Let .h∗ ∈ Λd be a solution of the dynamic user equilibrium problem with fixed demand. Then there exists multipliers .π ∈ Rn×|P | and .μ ∈ R|W | such

3.5 Kuhn-Tucker Conditions for DUE Problems

117

that Ψ (h∗ ) − π + Aμ = 0.

(3.59)

.

πpk h∗,k p = 0 ∀k ∈ {1, . . . , n}, p ∈ P. π ≥ 0

(3.60) (3.61)

 ∗,k  n×|P | ∈ R++ is the vector of discrete-time path effective Ψp

where .Ψ (h∗ ) = delays, and

A =

.

  u1 u2 · · · u|W | ∈ Rn|P |×|W |

(3.62)

where each .ui is a vector of ones and zeros such that uTi h =

n  

.

hkp

∀i ∈ {1, . . . , |W|}

(3.63)

p∈Pwi k=1

On the other hand, given any .h∗ ∈ Λd such that (3.59)-(3.61) hold, .h∗ must be a solution of the dynamic user equilibrium problem. Proof Define f : Rn×|P | −→ R|W | ,

.

g : Rn×|P | −→ Rn×|P | ,

f (h) = AT h − Q/Δt g(h) = − h

where .Q = (Qwi ) is the vector of travel demands. Then it is straightforward to verify that " # .Λd = h ∈ Rn×|P | : f (h) = 0, g(h) ≤ 0 According to Theorems 2.6 and 3.4, there exist .π ∈ Rn×|P | and .μ ∈ R|W | such that Ψ (h∗ ) + ∇g(h∗ ) π + ∇f (h∗ ) μ = 0

.

πi gi (h∗ ) = 0 ∀i ∈ {1, . . . , n|P|} π ≥ 0 Clearly, .∇g(h∗ ) = −I , where I denotes the identity matrix; and .∇f (h∗ ) = A. Identities (3.59)–(3.61) then follow immediately. Finally, since .f (·) is linear and .g(·) is convex, the K-T conditions (3.59)–(3.61) are also sufficient for optimality, as asserted by Theorem 2.7. 

118

3 The Variational Inequality Formulation of Dynamic User Equilibria

3.5.1.2

An Equivalent Linear System

We notice that the K-T conditions (3.59)–(3.61) are equivalent to the following system of linear equations. ⎡

−I

.

⎣

D

⎡ ⎤⎡ ⎤ ⎤ −Ψ (h∗ ) π A ⎦⎣ ⎦ = ⎣ ⎦ μ 0 0

(3.64)

where I is the .n|P| × n|P| identity matrix; A is given by (3.62) and (3.63); .D = diag(h∗ ) is an .n|P| × n|P| diagonal matrix whose diagonal elements are given by ∗ .h . Upon solving this linear system, one is prompted to verify if all elements of .π are non-negative. We note that the size of the system (3.64) will become very large as the number of paths or the number of time intervals increase; thus handling such a linear system may be a computationally intensive task. In the next proposition, we propose a way of decomposing the above linear system by origin-destination pairs. Proposition 3.1 Given each .i ∈ {1, . . . , |W|}, denote by .Ψ i (h∗ ) and .π i the sub-vectors of .Ψ (h∗ ) and .π , respectively, formed by entries associated with origindestination pair .wi . Let .μi be the .i th element of .μ. Denote by .I i the identity matrix with size .n|Pwi |. In addition, let .D i = diag(h∗,i ) where .h∗,i is the sub-vector of ∗ .h formed by entries associated with the origin-destination pair .wi . Then the linear system (3.64) is equivalent to a set of smaller systems: ⎡

−I i 1i

.

⎣

Di

0

⎤⎡

πi

⎦⎣

μi

⎡

⎤

⎦ = ⎣

⎦

⎤

−Ψ i (h∗ )

i ∈ {1, . . . , |W|}

(3.65)

0

where .1i is the column vector of one’s of size .n|Pwi |. Proof The conclusion can be shown via simple block matrix arithmetic.



The equivalent form (3.65) of the Kuhn-Tucker conditions will be applied to a DUE problem with fixed demand later in Sect. 9.1.2.

3.5.2 Application to DUE with Elastic Demand This section is concerned with elastic demand dynamic user equilibrium (E-DUE). In particular, the variational inequality formulation of E-DUE presented in Sect. 3.3 admits a set of Kuhn-Tucker necessary conditions that are easily expressible as a system of linear equations.

3.5 Kuhn-Tucker Conditions for DUE Problems

3.5.2.1

119

Kuhn-Tucker Conditions for Discrete-time DUE Problems with Elastic Demand

Let us again consider the following time grid with step size .Δt: t0 = t1 < . . . < tk−1 < tk < tk+1 < tn = tf

.

Using the same notation introduced in Sect. 3.5.1, we can express the discrete-time  defined in (3.30) as counterpart of .Λ d = Λ

.

⎧ ⎨ ⎩

  n×|P |+|W | X ≡ h, Q ∈ R+ : h ≥ 0,



Δt

p∈Pw

n 

hkp = Qw

∀w ∈ W

k=1

⎫ ⎬ ⎭

(3.66) n×|P |

where .hkp is the discrete value of path flow .hp (·) at .tk ; the vector .h ∈ R+ is the concatenation of .|P| vectors .hp1 , . . . , hp|P| where each .hpi ∈ Rn+ ; w denotes an origin-destination pair; and Q is the vector of realized demands between each .w ∈ W. Define the mapping .Ψ among finite-dimensional vector spaces: n×|P |

Ψ : R++

.

  h → Ψpk : p ∈ P, k = 1, . . . , n

n×|P |

→ R++ ,

(3.67)

where .Ψpk denotes the effective delay of drivers who depart at time .tk and travel along path p. Define the mapping .F as follows: n×|P |+|W |

F : R+

.

→ Rn×|P |+|W | :

   h, Q) → Ψ (h), −Θ(Q)

(3.68)

The inverse demand function for every .w ∈ W is vw = Θw [Q]

.

and we define   Θ = Θw : w ∈ W

.

We have the following theorem regarding the variational inequality formulation of discrete-time DUE with elastic demand, which is viewed as the discrete-time version of Theorem 3.2:

120

3 The Variational Inequality Formulation of Dynamic User Equilibria

Theorem 3.6 (Discrete-Time E-DUE Equivalent to a Finite-Dimensional VI) The simultaneous route-and-departure-time dynamic user equilibrium with elastic demand expressed in discrete time is equivalent to the following finite-dimensional variational inequality problem:   d such that find X∗ ≡ h∗ , Q∗ ∈ Λ .

 T F(X∗ ) (X − X∗ ) ≥ 0

⎫ ⎪ ⎬

⎪ d ⎭ ∀X ∈ Λ

  d , t0 , tf V Id Ψ, Λ

(3.69)

Theorem 3.7 (Sufficient and Necessary KT Conditions for DUE with Elastic  d be a solution of the dynamic user Demand) Let the pair . h∗ , Q∗ ∈ Λ equilibrium problem with elastic demand. Then there exist multipliers .π e ∈ n×|P |+|W | and .μe ∈ R|W | such that .R .

.

Ψ (h∗ )

− π e + Ae μe = 0.

− Θ[Q∗ ]

n×|P |

where .Ψ (h∗ ) ∈ R++

πie h∗i = 0

∀i ∈ {1, . . . , n × |P|}.

πie Q∗i = 0

∀i ∈ {n × |P| + 1, . . . , n × |P| + |W|}. (3.72)

πe ≥ 0

(3.73)

(3.71)

is the vector of discrete-time effective path delays, and

Ae =

.

(3.70)

    n|P |+|W ×|W | u1 u2 . . . u|W | ∈ R

(3.74)

where each .ui is a column vector such that T .ui

- . h Q

=

n   p∈Pwi k=1

hkp −

Qwi Δt

∀i ∈ {1, . . . , |W|}

(3.75)

 ∗ ∗  On  ∗ the ∗other hand, given any pair . h , Q ) ∈ Λd such that (3.70)-(3.73) hold, . h , Q ) must yield a dynamic user equilibrium with elastic demand. Proof We begin with rewriting the constraints expressed in (3.66) in standard form. That is, we define f e : Rn×|P |+|W | −→ R|W | ,

.

g e : Rn×|P |+|W | −→ Rn×|P | ,

f e (X) = [Ae ]T X g e (X) = − X

3.5 Kuhn-Tucker Conditions for DUE Problems

121

Then d = Λ

.

" n×|P |+|W | X ∈ R+ : f e (X) = 0,

# g e (X) ≤ 0

We now compute the gradients of .f e (·) and .g e (·): ∇f e (X) = Ae ,

∇g e (X) = − I

.

where I is the identity matrix of dimension .n × |P| + |W|. Finally, by Theorem 2.6 n|P |+|W | and Theorem 3.6, there exist .π e ∈ R+ and .μe ∈ Rn|P |+|W | such that F(x ∗ ) + [∇g e (x ∗ )]T π e + [∇f e (x ∗ )]T μe = 0.

(3.76)

.

πie gie (x ∗ ) = 0 ∀i ∈ {1, . . . , n|P| + |W|}. (3.77) πe ≥ 0

(3.78)

which gives (3.70)–(3.73). Since .f e (·) is linear and .g e (·) is convex, the K-T conditions (3.70)–(3.73) are sufficient for optimality. 

3.5.2.2

An Equivalent Linear System

Note that the K-T conditions (3.70)–(3.72) are equivalent to the following system of linear equations. ⎡

−I

.

⎢ ⎢ ⎣

De

Ae

0

⎤⎡

πe

⎥⎢ ⎥⎢ ⎦⎣

μe

⎤ −Ψ (h∗ ) ∗ ⎥ ⎥ ⎢ ⎥ = ⎢ Θ[Q ] ⎥ ⎦ ⎦ ⎣ ⎤

⎡

(3.79)

0

where I is the identity matrix with dimension .n|P| + |W|; .Ae is defined by (3.74) and (3.75). .D e denotes the diagonal matrix whose diagonal elements are given by the vector .(h∗ , Q∗ ) ∈ Rn|P |+|W | . Upon solving this linear system, one is prompted to verify if all elements of .π e are non-negative. Similarly to the case of fixed demand, the size of such a linear system blows up when the network size increases. Proposition 3.2 Given each .i ∈ {1, . . . , |W|}, let .Ψ i (h∗ ), π e,i and .μe,i be the sub-vectors of .Ψ (h∗ ), π e and .μe formed by elements associated with the origindestination pair .wi , respectively. In addition, let .I i be the identity matrix with size e,i = diag(h∗,i , Q∗ ) where .h∗,i is the sub-vector of .h∗ formed .n|Pwi | + 1. Let .D wi

122

3 The Variational Inequality Formulation of Dynamic User Equilibria

by entries associated with the origin-destination pair .wi , and .Q∗wi is the realized travel demand between origin-destination pair .wi . Finally, define Ae,i =

.



1 T 1, . . . , 1, − ∈ Rn|Pwi |+1 1 23 4 Δt n|Pwi |

Then the linear system (3.79) is equivalent to a set of smaller systems: ⎡ .

⎢ ⎢ ⎣

−I i

D e,i

Ae,i

0

⎤⎡

π e,i

⎥⎢ ⎥⎢ ⎦⎣

⎤ −Ψ i (h∗ ) ∗ ⎥ ⎥ ⎢ ⎥ = ⎢Θwi (Q )⎥ ⎦ ⎦ ⎣ ⎤

μe,i

⎡

(3.80)

0

Proof The decomposition follows from straightforward block matrix arithmetic.  The equivalent form (3.80) of the Kuhn-Tucker conditions will be applied to a DUE problem with elastic demand later in Sect. 9.1.4.

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Chapter 4

The Differential Variational Inequality Formulation of Dynamic User Equilibria

In this chapter, we present the differential variational inequality (DVI) formalism and use it to relate the much studied calculus of variations, including the theory of optimal control, to differential Nash games. The games we consider are motivated by the notion of dynamic user equilibrium as presented in the previous chapter. A good review of recent insights into abstract differential variational inequality theory, including computational methods for solving such problems, is provided by Pang and Stewart (2008). Also, differential variational inequalities involving the kind of explicit, agent-specific control variables employed herein are presented in Friesz (2010). An introduction to the mathematical foundations of DVI theory has been provided earlier, in Sect. 2.5. The main purpose of this chapter is to articulate the relationship between dynamic user equilibria and DVI. In particular, we will show that the DUE problem with fixed demand, elastic demand, and bounded rationality can all be formulated as differential variational inequalities. This is done by using the minimum principle from optimal control theory. The DVI formulations achieved here are significant because they allow the still emerging theory of differential variational inequalities to be employed for the analysis and computation of solutions of these DUE problems, when simultaneous departure time and route choice are within the purview of users, all of which constitutes a foundation problem within the field of dynamic traffic assignment. It is well known that a variational inequality is equivalent to a fixed-point problem under very mild regularity conditions in both finite-dimensional and infinitedimensional spaces. As such, an appropriately defined fixed-point algorithm is an obvious choice of numerical solution scheme for DVIs. In this chapter, we provide an explanation and numerical examples of a specific type of fixed-point algorithm. We also present a self-adaptive projection algorithm, as well as a proximal-point algorithm.

© Springer Nature Switzerland AG 2022 T. L. Friesz, K. Han, Dynamic Network User Equilibrium, Complex Networks and Dynamic Systems 5, https://doi.org/10.1007/978-3-031-25564-9_4

125

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4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

4.1 DVI Formulation of DUE with Fixed Demand In the previous chapter, the simultaneous route-and-departure-time (SRDT) choice dynamic user equilibrium (DUE) with fixed demand is related to an infinitedimensional variational inequality via a measure theoretic argument. The main result is that .h∗ ∈ Λ is an SRDT DUE if and only if it solves the following VI in an Hilbert space:

.

find h∗ ∈ Λ such that

⎫ ⎬



⎭

Ψ (h∗ ),

h − h∗



≥ 0

∀h ∈ Λ

  V I Ψ, Λ, [t0 , tf ]

(4.1)

where the set of feasible path flows reads Λ =

.

⎧ ⎨ ⎩



h≥0:

p∈Pij

tf

hp (t) dt = Qij

∀ (i, j ) ∈ W

t0

⎫ ⎬ ⎭

  |P | ⊆ L2+ t0 , tf (4.2)

In this section, we will present another formulation of the SRDT DUE with fixed demand, which is closely related to optimal control theory. In this formulation the trip matrix is assumed to be given and fixed. It was first noted in Friesz et al. (2011) that (4.1) is equivalent to a differential variational inequality (DVI). This is most easily seen by noting that the flow conservation constraints expressed in (4.2) may be restated as ⎫ dyij (t) ⎪ = hp (t) ∀(i, j ) ∈ Pij ⎪ ⎪ ⎬ dt .

p∈Pij

yij (t0 ) = 0 yij (tf ) = Qij

∀(i, j ) ∈ Pij ∀(i, j ) ∈ Pij

⎪ ⎪ ⎪ ⎭

which is recognized as a two point boundary value problem. Here, .yij (t) is interpreted as cumulative departures associated with O-D pair .(i, j ) by time .t ∈ [t0 , tf ]. It is treated as the state variable of the differential system associated with the DVI; see Sect. 2.4. Accordingly, the following set of feasible path departure rates is obviously equivalent to (4.2). . .Λ1 =

⎧ ⎨

dyij = h≥0: hp (t), yij (t0 ) = 0, yij (tf ) = Qij ⎩ dt p∈Pij

∀(i, j ) ∈ W

⎫ ⎬ ⎭

(4.3)

4.1 DVI Formulation of DUE with Fixed Demand

127

As a consequence (4.1) may be conceptually expressed as a differential variational inequality: ⎫ ⎬

find h∗ ∈ Λ1 such that .



Ψ (h∗ ), h − h

 ∗

≥ 0

∀h ∈ Λ1

⎭

  DV I Ψ, Λ1 , [t0 , tf ]

(4.4)

The equivalence between the DVI (4.4) and the original DUE problem is established in the theorem below. Theorem 4.1 (DUE with Fixed Demand Equivalent to a Differential Variational Inequality) Assume .Ψp (·, h) : [t0 , tf ] → R+ is measurable and strictly positive for all .p ∈ P and all .h ∈ Λ1 . A vector of path departure rates .h∗ ∈ Λ1 is a  ∗ DUE with fixed demand if and only if .h solves .DV I Ψ, Λ1 , [t0 , tf ] , as defined by (4.4). Proof We follow the proof of Friesz et al. (2011) and use the necessary conditions of optimal control problem.  The measurability  assumption assures that the integrals used in articulating .DV I Ψ, Λ1 , [t0 , tf ] are well-defined. The proof is in two parts. (i)[Necessity] Given that .h∗ is a dynamic user equilibrium, we have that Ψp (t, h∗ ) ≥ vij∗

.

∀(i, j ) ∈ W, p ∈ Pij , t ∈ [t0 , tf ]

(4.5)

Therefore if .hp (t) − h∗p (t) ≥ 0 we have Ψp (t, h∗ )(hp (t) − h∗p (t)) ≥ vij∗ (hp (t) − h∗p (t))

.

∀(i, j ) ∈ W, p ∈ Pij , t ∈ [t0 , tf ]

(4.6)

However hp (t) − h∗p (t) < 0 ⇒ h∗p (t) > hp (t) ≥ 0 ⇒ h∗p (t) > 0

.

which requires (4.5) to hold as an equality, thereby assuring (4.6) is valid for any hp ∈ Λ. As a consequence, we may sum and integrate both sides of (4.6) to obtain

.



tf

.

p∈Pij

t0

Ψp (t, h∗ )(hp (t) − h∗p (t)) dt ≥

p∈Pij

=

tf t0

(i, j )∈W

=



(i, j )∈W

vij∗ (hp (t) − h∗p (t)) dt

vij∗

p∈Pij

tf

t0

hp (t) − h∗p (t) dt

vij∗ (Qij − Qij ) = 0

128

4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

This completes the “only if” part. (ii) [Sufficiency] The differential variational inequality of interest may be written as tf tf ∗ . Ψp (t, h )hp (t) dt ≥ Ψp (t, h∗ )h∗p (t) dt (i, j )∈W p∈Pij

t0

(i, j )∈W p∈Pij

t0

∀h ∈ Λ1 which means that the solution .h∗ ∈ Λ1 satisfies the optimal control problem .



min J0 =

vij∗ [Qij − yij (tf )] +

(i, j )∈W





tf

Ψp (t, h∗ )hp (t) dt

t0

(i, j )∈W p∈Pij

(4.7) subject to .



dyij = dt

hp (t)

∀(i, j ) ∈ W.

(4.8)

p∈Pij

y0 (t0 ) = 0

∀(i, j ) ∈ W.

(4.9)

h ≥ 0

(4.10)

where the .vij∗ are presently dual variables for the terminal conditions on the state variables. The Hamiltonian for problem (4.7)-(4.10) is

H =



.

(i, j )∈W p∈Pij

⎧ ⎨

=

(i, j )∈W

⎩



Ψp (t, h∗ )hp +

λij

(i, j )∈W

(Ψp (t, h∗ ) + λij )hp

p∈Pij

⎫ ⎬



hp

p∈Pij

⎭

(4.11)

where the adjoint equations are .

dλij ∂H = 0 = − dt ∂yij

∀(i, j ) ∈ W, p ∈ Pij , t ∈ [t0 , tf ]

(4.12)

with transversality conditions λij (t0 ) =

.

∂



(i, j )∈W

vij∗ [Qij − yij (t0 )]

∂yij (tf )

= − vij∗ = constant

∀(i, j ) ∈ W, p ∈ Pij , t ∈ [t0 , tf ]

(4.13)

4.2 Fixed-Point Problem Formulation of DUE with Fixed Demand

129

Putting together, (4.12) and (4.13) imply that λij (t) ≡ − vij∗

∀(i, j ) ∈ W, t ∈ [t0 , tf ]

.

According to the Pontryagin minimum principle, the controls must obey, for each instance of time, the following .

min H

such that

−h ≤ 0

for which the Kuhn-Tucker conditions are Ψp (t, h∗ ) − vij∗ = μp (t) ≥ 0

.

∀(i, j ) ∈ W, p ∈ Pij , t ∈ [t0 , tf ] (4.14)

where the .μp are dual variables satisfying the complementarity slackness conditions μp (t) hp (t) = 0

∀(i, j ) ∈ W, p ∈ Pij , t ∈ [t0 , tf ]

.

(4.15)

From (4.14) and (4.15) we immediately have the conditions of SRDT DUE, namely h∗p (t) > 0, p ∈ Pij ⇒ Ψp (t, h∗ ) = vij∗

.

Ψp (t, h∗ ) > vij∗ , p ∈ Pij ⇒ h∗p (t) = 0 with the obvious interpretation that each dual variable .vij∗ is the essential infimum of the effective unit path delay .Ψp (t, h∗ ). Thus we are assured that any solution of our differential variational inequality is a dynamic user equilibrium relative to path and departure time choice. 

4.2 Fixed-Point Problem Formulation of DUE with Fixed Demand It should come as no surprise that there is often an equivalent functional fixedpoint problem corresponding to a given differential Nash game. This formulation provides  an immediate, simple, and sometimes quite effective algorithm for solving .DV I Ψ, Λ1 , [t0 , tf ] , as we shall see later in Chap. 6. Theorem 4.2 (Fixed-Point Problem Equivalent to DUE) Assume that .Ψp (·, h) : [t0 , tf ] → R+ is measurable for all .p ∈ P and .h ∈ Λ1 . Then the fixed-point problem   h∗ = PΛ1 h∗ − αΨ (t, h∗ )

.

(4.16)

130

4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

is equivalent to .DV I (Ψ, Λ1 , [t0 , tf ]) where .PΛ1 [·] is the minimum norm projection onto .Λ1 and .α ∈ R++ . Proof The fixed-point problem considered requires that 

 1 h∗ − αΨ (t, h∗ ) − h2 : h ∈ Λ1 .h = argmin 2 h ∗

 (4.17)

That is, we seek the solution of the optimal control problem

.



min J (h) = h

  μij Qij − y(tf )

(i, j )∈W

+

tf t0

1 2

 2 h∗ (t) − αΨ (t, h∗ ) − h(t) dt



(4.18)

(i, j )∈W p∈Pij

subject to .

dyij (t) = dt



hp (t)

∀(i, j ) ∈ W.

(4.19)

p∈Pij

yij (t0 ) = 0

∀(i, j ) ∈ W.

hp (t) ≥ 0

∀t ∈ [t0 , tf ],

(4.20) ∀p ∈ P

(4.21)

This problem has the Hamiltonian H =

.

1 2



2  h∗p (t) − αΨp (t, h∗ ) − hp (t) +

(i, j )∈W p∈Pij



λij (t)

(i, j )∈W



hp (t)

p∈Pij

(4.22) which is convex in its controls h and has no state dependence, so the minimum principle and supporting optimality conditions are necessary and sufficient. Hence, it will be enough to show that any solutions of (4.18)–(4.21) satisfy .DV I (Ψ, Λ1 , [t0 , tf ]). The minimum principle for (4.22) has the Kuhn-Tucker conditions   ∗ ∗ ∗∗ · (−1) + λij = ρp . hp − αΨp (t, h ) − hp ∀(i, j ) ∈ W, p ∈ Pij . (4.23) ρp h∗∗ p = 0

∀(i, j ) ∈ W,

p ∈ Pij . (4.24)

ρp ≥ 0

∀(i, j ) ∈ W,

p ∈ Pij (4.25)

4.3 DVI Formulation of DUE with Elastic Demand

131

where .h∗∗ denotes the optimal solution of .

min H

such that h ≥ 0

By virtue of (4.17), .h∗ = h∗∗ ; therefore, we may restate (4.23) as αΨp (t, h∗ ) + λij = ρp

∀(i, j ) ∈ W,

.

p ∈ Pij

(4.26)

Note also that the adjoint equations and associated transversality conditions are .

dλij ∂H = (−1) = 0 dt ∂yij

∀(i, j ) ∈ W

∂μij [Qij − yij (tf )] = − μij ∂yij (tf )

λij (tf ) =

∀(i, j ) ∈ W

Consequently λij = − μij

.

∀(i, j ) ∈ W,

t ∈ [t0 , tf ]

(4.27)

Because .h∗ = h∗∗ , we have by (4.24) that h∗p > 0 ⇒ ρp = 0

.

∀p ∈ Pij

hence h∗p (t) > 0, p ∈ Pij ⇒ αΨp (t, h∗ ) = −λij = μij ⇒ Ψp (t, h∗ )

.

=

μij . = vij α

(4.28)

Furthermore, by (4.25), (4.26) and (4.27), Ψp (t, h∗ ) ≥ −

.

μij λij = = vij α α

(4.29)

Expressions (4.28) and (4.29) in combination are recognized as the essential feature of .DU E(Ψ, Λ, [t0 , tf ]). 

4.3 DVI Formulation of DUE with Elastic Demand The DVI formalism can also accommodate dynamic user equilibrium with endogenous (elastic) travel demand. Such a DVI formulation for the E-DUE problem is not a straightforward extension. In particular, this DVI has both infinite-dimensional

132

4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

and finite-dimensional terms (Friesz and Meimand, 2014). Moreover, for any given origin-destination pair, inverse travel demand corresponding to a dynamic user equilibrium depends on the terminal value of a state variable representing cumulative departures. The DVI formulation presented here is significant because it allows the still emerging theory of differential variational inequalities to be employed for the analysis and computation of solutions of the elastic-demand DUE problem when simultaneous departure time and route choices are within the purview of users. We continue to use the same concepts and notation introduced in Sect. 3.3.1 and recap some of the key variables and recap demand case, the  here. In the elastic  endogenous travel demand matrix .Q = Qij : (i, j ) ∈ W is related to the O-D minimum travel cost .vij via the inverse demand function: for every .(i, j ) ∈ W, vij = Θij [Q]

(4.30)

.

We introduce the following dynamics: .

dyij (t) = dt



hp (t),

yij (t0 ) = 0

∀ (i, j ) ∈ W

(4.31)

p∈Pij

where .yij (t) is treated as the cumulative departure for O-D pair .(i, j ) by time t; and the realized total demand is obviously .yij (tf ), which is determined endogenously. As a consequence, we employ the following alternative form of the feasible set: ⎧ ⎨

dyij (t) = hp (t) , yij (t0 ) = 0, .Λ0 = h≥0: ⎩ dt p∈Pij

  |P | ⊆ L2+ t0 , tf

∀(i, j ) ∈ W

⎫ ⎬ ⎭ (4.32)

Note that the feasible set .Λ0 in (4.32) is expressed as a set of path departure rate vectors h, since knowledge of h completely determines the demands that satisfy the initial value problem (4.31). Finally, the inverse demand function (4.30) can be equivalently written as   vij = Θij y(tf )

.

∀(i, j ) ∈ W

where  .  y(·) . = yij (·) : (i, j ) ∈ W Experience with differential games in continuous time suggests that an elastic demand dynamic user equilibrium is equivalent to the variational inequality presented in the following theorem, under suitable regularity conditions.

4.3 DVI Formulation of DUE with Elastic Demand

133

Theorem 4.3 (E-DUE to a Differential Variational Inequality)   Equivalent Assume .Ψp (·, h) : t0 , tf −→ R1++ is measurable and strictly positive for all .p ∈ P and all .h ∈ Λ0 . Also assume that the elastic travel demand function is invertible, with inverse .Θij [Q] for all .(i, j ) ∈ W. A vector of departure rates (path flows) .h∗ ∈ Λ0 is a dynamic user equilibrium with the associated equilibrium terminal demand .y ∗ (tf ) if and only if .h∗ solves .

p∈P

tf t0

⎫ find h∗ ∈ Λ0 such that ⎪ ⎪ ⎪   ⎬   ∗ ∗ ∗ ∗ Ψp (t, h )(hp − hp )dt − Θij y (tf ) yij (tf ) − yij (tf ) ≥ 0 ⎪ ⎪ (i, j )∈W ⎪ ⎭ ∀h ∈ Λ0 (4.33)

  We call the DVI problem in (4.33) .DV I Ψ, Θ, [t0 , tf ] .

  Proof (i) [Sufficiency] We note that the problem .DV I Ψ, Θ, [t0 , tf ] may be written as: find .h∗ ∈ Λ0 such that



tf

Ψp (t, h∗ )hp (t) dt −

.

(i,j )∈W p∈Pij

≥



t0

   Θij y ∗ tf yij (tf )

(i,j )∈W



tf

t0

(i,j )∈W p∈Pij





Ψp (t, h∗ )h∗p (t) dt −

  Θij y ∗ (tf ) yij∗ (tf )

(4.34)

(i,j )∈W

for all .h ∈ Λ0 . Inequality (4.34) means that the solution .h∗ ∈ Λ0 satisfies the optimal control problem .

min J0 = −



  Θij y ∗ (tf ) yij (tf ) +

(i,j )∈W



(i,j )∈W p∈Pij

tf

Ψp (t, h∗ )hp (t) dt

t0

(4.35) subject to .

dyij (t) = dt



hp (t)

∀(i, j ) ∈ W.

(4.36)

p∈Pij

yij (t0 ) = 0 h ≥ 0

∀(i, j ) ∈ W.

(4.37) (4.38)

This optimal control problem may not be used for computation, since it involves knowledge of the DVI solution. However, it may  be used to express necessary and sufficient conditions for the solution of .DV I Ψ, Θ, [t0 , tf ] . In particular, the

134

4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

  Hamiltonian for .DV I Ψ, Θ, [t0 , tf ] is



H =

.

(i,j )∈W p∈Pij

⎧ ⎨ 

=

(i,j )∈W

⎩



Ψp (t, h∗ ) hp +

λij

(i,j )∈W



Ψp (t, h∗ ) + λij hp

p∈Pij

⎫ ⎬



hp

p∈Pij

⎭

where the adjoint equations are .

dλij ∂H = 0 = − dt ∂yij

∀ (i, j ) ∈ W,

p ∈ Pij ,

  t ∈ t0 , tf

(4.39)

with transversality conditions ∂   λij tf = −



(i,j )∈W

   Θij y ∗ tf yij (tf ) ∂yij (tf )

∀ (i, j ) ∈ W,

p ∈ Pij ,

   = − Θij y ∗ tf

  t ∈ t0 , tf

(4.40)

It is clear from (4.39) and (4.40) that    λij (t) = − Θij y ∗ tf , a constant

.

We note that the Hamiltonian is linear in h and does not depend explicitly on the state variables. By Theorem 3.7 of Friesz (2010), the Mangasarian sufficiency theorem assures the minimum principle and associated necessary conditions are also sufficient. Since h is a control vector and must obey the minimum principle in .R|P | for each instant of time, we enforce .

min H h

such that

−h≤0

for which the Kuhn-Tucker conditions are    = ρp ≥ 0 Ψp (t, h∗ ) − Θij y ∗ tf

.

  ∀ (i, j ) ∈ W, p ∈ Pij , t ∈ t0 , tf (4.41)

where the .ρp are dual variables satisfying the complementary slackness conditions ρp h∗p = 0

.

∀(i, j ) ∈ W,

p ∈ Pij ,

t ∈ [t0 , tf ]

(4.42)

4.3 DVI Formulation of DUE with Elastic Demand

135

From (4.41) and (4.42) we have immediately h∗p > 0, p ∈ Pij ⇒ Ψp (t, h∗ ) = Θij [y ∗ (tf )]

.

Ψp (t, h∗ ) > Θij [y ∗ (tf )], p ∈ Pij ⇒ h∗p = 0

.

which are recognized as conditions describing a dynamic user equilibrium. (ii) [Necessity] We have noted above that (4.41) is equivalent to a dynamic user equilibrium, thus showing that (4.41) corresponds to a solution of .DV I Ψ, Θ, [t0 , tf ] and will complete the demonstration that .DV I Ψ, Θ, [t0 , tf ] is equivalent to a dynamic user equilibrium. In particular note that ρp (hp − h∗p ) = ρp hp − ρp h∗p = ρp hp ≥ 0

.

  so that .∀ (i, j ) ∈ W, p ∈ Pij , t ∈ t0 , tf .

  Ψp (t, h∗ ) − Θij [y ∗ (tf )] (hp − h∗p ) ≥ 0

(4.43)

where .h, h∗ ∈ Λ0 . From (4.43) we obtain    Ψp (t, h∗ )(hp − h∗p ) − Θij y ∗ tf (hp − h∗p ) ≥ 0

.

(4.44)

From (4.44), we obtain

tf

.

p∈P

t0

Ψp (t, h∗ )(hp − h∗p )dt



−

   Θij y ∗ tf

⎛ tf

⎝

t0

(i,j )∈W

p∈Pij

hp −



⎞ h∗p ⎠ dt ≥ 0

(4.45)

p∈Pij

Therefore

tf

.

p∈P

−

t0

Ψp (t, h∗ )(hp (t) − h∗p (t)) dt

(i,j )∈W

  Θij y ∗ (tf )

tf

t0



∗ dyij (t) dyij (t) − dt dt

 dt ≥ 0

(4.46)

  from which .DV I Ψ, Θ, [t0 , tf ] is obtained immediately, since .yij (t0 ) = yij∗ (t0 ) = 0. 

136

4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

4.4 Fixed-Point Problem Formulation of DUE with Elastic Demand The fixed-point problem (FPP) reformulation of the E-DUE problem relies on a projection operator to be defined on an extended Hilbert space. To articulate such . a projection and the accompanying space, we introduce the product space .E =  2 |P | L [t0 , tf ] × R|W | , which is a space with the natural inner product defined as follows |P | . X, Y E = .

tf

ξi (t) · ηi (t) dt +

t0

|W |

uj vj

j =1

  X = ξ1 (·), . . . , ξ|P | (·), u1 , . . . , u|W | ∈ E   Y = η1 (·), . . . , η|P | (·), v1 , . . . , v|W | ∈ E i=1

(4.47)

The inner product . ·, · E induces the following norm on the space E 1/2

XE = X , X E

.

(4.48)

making E a metric space. In the following proposition, we show that the inner product . ·, · E and the norm . · E are well defined, and the resulting space E is indeed a Hilbert space. Proposition 4.1 The inner product . ·, · E and norm . · E are well defined. In addition, the space E, equipped with . ·, · E and the induced metric, is a Hilbert space over .R, the set of real numbers. Proof A well-defined inner product over .R must satisfy, for all .X, Y, Z ∈ E, 1. symmetry, i.e., . X, Y E = Y, X E ; 2. linearity, i.e., . aX, Y E = a X, Y E for all .a ∈ R, and . X + Y, Z E = X, Z + Y, Z E ; 3. positive-definiteness, i.e., . X, X E ≥ 0 and . X, X E = 0 ⇒ X = 0. It is straightforward to verify that the inner product defined in (4.47) satisfies all these three conditions and thus is well defined. Consequently, the induced norm  2 |P | . · E is also well defined. Finally, since both . L [t0 , tf ] and .R|W | are complete metric spaces, their product space E is also a complete metric space, and hence a Hilbert space. 

4.4 Fixed-Point Problem Formulation of DUE with Elastic Demand

137

 of admissible pairs .(h, Q), defined earlier in (3.30) and recapped here The set .Λ as = Λ

.

⎧ ⎨ ⎩

(h, Q) : h ≥ 0,

p∈Pij

tf

hp (t) dt = Qij

∀(i, j ) ∈ W

t0

⎫ ⎬ ⎭

 |P | ⊂ L2 [t0 , tf ] × R|W |

(4.49)

can now be embedded in the extended space E. In view of the inverse demand function .Θ = (Θij : (i, j ) ∈ W), we introduce the following notation. . Θ − = (−Θij : (i, j ) ∈ W) :

.

|W |

|W |

R+

−→ R−−

Consequently, we define the mapping  −→ E, F :Λ

(h, Q) →

.

  Ψ (h) , Θ − [Q]

(4.50)

  W|  .Ψ (h) ∈ L2+ [t0 , tf ] |W | , Θ − [Q] ∈ R|−− where .(h, Q) ∈ Λ, . Such a mapping is clearly well-defined. Note that the set (4.49) can be trivially re-written as a two-point boundary problem, leading to the following alternative definition of the feasible set.  dyij (t) |W | 2 |P |  hp (t), .Λ1 = (h, Q) ∈ (L+ [t0 , tf ]) × R+ : = dt  yij (t0 ) = 0, yij (tf ) = Qij

p∈Pij

∀(i, j ) ∈ W

(4.51)

 is a solution of the DUE problem with Recall from Sect. 3.3.1 that .(h∗ , Q∗ ) ∈ Λ elastic demand if and only if the following are satisfied: h∗p (t) > 0, p ∈ Pij ⇒ Ψp (t, h∗ ) = Θij [Q∗ ]

∀(i, j ) ∈ W

(4.52)

∀p ∈ Pij , ∀v (t) ∈ [t0 , tf ], ∀(i, j ) ∈ W

(4.53)

.

Ψp (t, h∗ ) ≥ Θij [Q∗ ]

.

Experience with optimal control theory suggests that (4.52) and (4.53) admit a fixed point problem representation, as stated and proved below. Theorem 4.4 (FPP Formulation of E-DUE) The fixed point problem   X∗ = PΛ1 X∗ − αF(X∗ )

.

(4.54)

138

4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

is equivalent to the DUE problem with elastic demand, where .α ∈ R++ ; .X∗ = |W | (h∗ , Q∗ ) ∈ (L2+ [t0 , tf ])|P | × R+ ; .PΛ1 [·] is the minimum norm projection onto 1 . .Λ Proof The fixed point problem (4.54) requires that X

.

∗



 1 X∗ − αF(X∗ ) − X2 : = arg min E X 2

1 X∈Λ

 (4.55)

By writing (4.55), we seek the solution of the optimal control problem

.

tf

min J (X) = X

t0

1 2

+

1 2

2  h∗p (t) − αΨp (t, h∗ ) − hp (t) dt



(i,j )∈W p∈Pij



2   yij∗ (tf ) − αΘij− y ∗ (tf ) − yij (tf )

(4.56)

(i,j )∈W

subject to .

dyij (t) = hp (t) dt

∀ (i, j ) ∈ W.

(4.57)

p∈Pij

yij (t0 ) = 0

∀ (i, j ) ∈ W.

(4.58)

h≥0

(4.59)

The Hamiltonian for the above optimal control problem is H =

.

1 2





h∗p (t) − αΨp (t, h∗ ) − hp (t)

(i,j )∈W p∈Pij

2

+



λij

(i,j )∈W



hp (t)

p∈Pij

where the adjoint equations are .

dλij ∂H = − = 0 dt ∂yij

∀(i, j ) ∈ W, p ∈ Pij , t ∈ [t0 , tf ]

(4.60)

with

λij (tf ) =

.

=

1∂ 2



 2   ∗ (t ) − αΘ − y ∗ (t ) − y (t ) y f f ij f (i,j )∈W ij ij ∂yij (tf )

  −yij∗ (tf ) + αΘij− y ∗ (tf ) + yij (tf )

  = αΘij− y ∗ (tf )

(4.61)

4.5 DVI Formulation of DUE with Bounded Rationality

139

By the minimum principle, we enforce .

min H h

s.t.

−h ≤ 0

for which the Kuhn-Tucker conditions are (we denote the Kuhn-Tucker point by h∗∗ )

.

.

  − h∗p (t) − αΨp (t, h∗ ) − h∗∗ p (t) + λij = ρp

∀(i, j ) ∈ W,

p ∈ Pij . (4.62)

ρp h∗∗ p = 0 ρp ≥ 0

∀(i, j ) ∈ W, ∀(i, j ) ∈ W,

p ∈ Pij .

(4.63)

p ∈ Pij

(4.64)

By virtue of (4.54), we have .h∗ = h∗∗ . Equations (4.60) and (4.61) then imply − ∗ .λij ≡ αΘ [y (tf )]. Therefore, we may restate (4.62) and (4.64) as ij αΨp (t, h∗ ) + αΘij− [Q∗ ] = ρp ≥ 0

.

∀(i, j ) ∈ W,

p ∈ Pij

. which implies that (recall that .Θ − = −Θ) Ψp (t, h∗ ) ≥ Θij [Q∗ ]

.

∀ν (t) ∈ [t0 , tf ],

∀(i, j ) ∈ W,

p ∈ Pij

Consequently, we have   h∗p (t) > 0 ⇒ ρp = 0 ⇒ Ψp (t, h∗ ) = Θij Q∗

.

∀t ∈ [t0 , tf ],

which are recognized as conditions describing a DUE with elastic demand.

p ∈ Pij 

4.5 DVI Formulation of DUE with Bounded Rationality The DVI formulation of dynamic user equilibrium with bounded user rationality is tremendously facilitated by the introduction of the new operators .φ ε or .Φ ε , defined earlier in Sect. 3.4.2. In the following sections, the DVI formulations of DUE with fixed and endogenous tolerances, respectively, will be established.

140

4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

4.5.1 With Exogenous Tolerances (BR-DUE) We assume that the travel demand is fixed, that is, we search for solutions of the BR-DUE problem with exogenous demand in the following feasible set: Λ =

.

⎧ ⎨ ⎩



h≥0:

p∈Pij

tf

hp (t) dt = Qij

∀ (i, j ) ∈ W

t0

⎫ ⎬ ⎭

  |P | ⊆ L2+ t0 , tf (4.65)

Let us introduce the function .yij (·) : [t0 , tf ] → [0, Qij ] for each origindestination pair .(i, j ), which represents the total traffic volume that has departed from origin i with the intent of reaching destination j . Note that the set of feasible path flows .Λ defined in (4.65) can be restated as Λ1 =

.

⎧ ⎨ ⎩

h≥0:

dyij (t) = hp (t), dt

yij (t0 ) = 0,

p∈Pij

yij (tf ) = Qij

∀(i, j ) ∈ W

(4.66)

Let us recall the definition of bounded rationality dynamic user equilibrium (BRDUE) with exogenous tolerances. Definition 4.1 (BR-DUE) Given the vector of constant tolerances .ε =   |W | εij : (i, j ) ∈ W ∈ R+ , a vector of departure rates .h∗ ∈ Λ1 is a BR-DUE associated with .ε if, for all .(i, j ) ∈ W, we have h∗p (t) > 0, p ∈ Pij ⇒ Ψp (t, h∗ ) ∈ [vij (h∗ ), vij (h∗ ) + εij ]

.

∀ν (t) ∈ [t0 , tf ]

(4.67)

where .vij (h∗ ) is the essential infimum of the effective path delays between origin  destination pair .(i, j ). We denote this equilibrium by BR-DUE. Ψ, ε, Λ1 , [t0 , tf ] .   |W | Given any vector .ε = εij : (i, j ) ∈ W ∈ R+ , we define the mapping  |P | φ ε : Λ1 → L2+ [t0 , tf ] ,

.

  h → φpε (·, h) : p ∈ P

(4.68)

where ! φpε (t, h) = max Ψp (t, h), vij (h) + εij

.

∀p ∈ Pij

∀(i, j ) ∈ W (4.69)

4.5 DVI Formulation of DUE with Bounded Rationality

141

and .vij (h) denotes the essential infimum of the effective delay between O-D pair (i, j ). We are now ready to state the DVI equivalence theorem for BR-DUE problems with exogenous tolerances.

.

Theorem 4.5 (DVI Formulation of BR-DUE with Exogenous Tolerances) |W | Given .ε ∈ R+ , assume .φ ε (·, h) is measurable and strictly positive for all ∗ ∈ Λ is a .p ∈ P and all .h ∈ Λ1 . A vector of departure rates (path flows) .h 1 ∗ bounded rationality dynamic user equilibrium if and only if .h solves the following differential variational inequality .

p∈P

tf t0

find h∗ ∈ Λ1 such that φpε (t, h∗ )(hp (t) − h∗p (t)) dt ≥ 0 ∀ h ∈ Λ1

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

  DV I φ ε , Λ1 , [t0 , tf ]

(4.70)

∗ Proof (i) [Sufficiency] Given  ε that .h ∈ Λ  1 is a solution of the differential variational inequality .DV I φ , Λ1 , [t0 , tf ] , we derive the following obvious relation: tf tf . φpε (t, h∗ )h∗p (t) dt ≤ φpε (t, h∗ )hp (t) dt p∈P

t0

p∈P

t0

for all .h ∈ Λ1 . In other words, .h∗ is the solution of the optimal control problem: .

min J (h) =





(i, j )∈W p∈Pij

tf t0



φpε (t, h∗ ) hp (t) dt +

  μij Qij − yij (tf )

(i, j )∈W

(4.71) subject to .

d yij (t) = dt



hp (t)

∀(i, j ) ∈ W.

(4.72)

p∈Pij

yij (t0 ) = 0

∀(i, j ) ∈ W.

(4.73)

h ≥ 0

(4.74)

where each .μij is the dual variable for the terminal conditions on .yij (·), .(i, j ) ∈ W. The Hamiltonian for problem (4.71)-(4.74) is H =





.

(i, j )∈W p∈Pij

φpε (t, h∗ ) hp +

(i, j )∈W

λij

p∈Pij

hp

142

4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

where the adjoint dynamics are .

d ∂H λij (t) = − = 0 dt ∂yij

∀(i, j ) ∈ W,

t ∈ [t0 , tf ],

(4.75)

and the transversality conditions read λij (tf ) =

∂

.



(i, j )∈W

  μij Qij − yij (tf )

∂yij (tf )

= −μij

∀(i, j ) ∈ W,

t ∈ [t0 , tf ] (4.76)

Taken together, (4.75) and (4.76) imply that λij (t) ≡ − μij

∀(i, j ) ∈ W,

.

∀t ∈ [t0 , tf ]

(4.77)

According to the minimum principle, we have h∗ = arg min H

.

h

such that − h ≤ 0

for which the Kuhn-Tucker conditions are φpε (t, h∗ ) + λij = ρp ≥ 0

∀p ∈ Pij ,

∀(i, j ) ∈ W,

∀ν (t) ∈ [t0 , tf ]. (4.78)

ρp (t) h∗p (t) = 0

∀p ∈ Pij ,

∀(i, j ) ∈ W,

∀ν (t) ∈ [t0 , tf ]. (4.79)

ρp (t) ≥ 0

∀p ∈ Pij ,

∀(i, j ) ∈ W,

∀ν (t) ∈ [t0 , tf ] (4.80)

.

where .ρp are dual variables for the non-negativity constraints. From (4.78) and (4.79) we have for all .(i, j ) ∈ W that h∗p (t) > 0, p ∈ Pij ⇒ φpε (t, h∗ ) = − λij = μij

∀ν (t) ∈ [t0 , tf ]. (4.81)

φpε (t, h∗ ) = ρp − λij = ρp + μij ≥ μij

∀ν (t) ∈ [t0 , tf ] (4.82)

.

from which we conclude that ! μij = essinf φpε (t, h∗ ) : t ∈ [t0 , tf ], p ∈ Pij

.

∀(i, j ) ∈ W

4.5 DVI Formulation of DUE with Bounded Rationality

143

In view of the definition of .φpε (4.69), we restate (4.81) as h∗p (t) > 0, p ∈ Pij ⇒ Ψ (t, h∗ ) ≤ vij (h∗ )+εij

.

∀ν (t) ∈ [t0 , tf ],

∀(i, j ) ∈ W,

which are recognized as conditions describing a dynamic user equilibrium with bounded rationality, where .vij (h∗ ) is the essential infimum of effective path delays between origin-destination .(i, j ). (ii)[Necessity] Assume .h∗ ∈ Λ1 yields a BR-DUE flow. For each origin-destination pair .(i, j ) ∈ W, define .μεij (h) to be the essential infimum of .φ ε (h) within O-D pair .(i, j ); that is, ! . μεij (h) = essinf φpε (t, h) : t ∈ [t0 , tf ], p ∈ Pij

.

It is then easy to verify that .μεij (h) = vij (h) + εij for all .(i, j ) ∈ W and for all ∗ .h ∈ Λ1 . For a BR-DUE solution .h , we have for all .(i, j ) ∈ W that h∗p (t) > 0, p ∈ Pij ⇒ vij (h∗ ) ≤ Ψp (t, h∗ ) ≤ vij (h∗ ) + εij ,

.

which translates to h∗p (t) > 0, p ∈ Pij ⇒ φpε (t, h∗ ) = μεij (h∗ )

∀(i, j ) ∈ W

.

Therefore, given any .h ∈ Λ1 , we have



tf

.

(i, j )∈W p∈Pij

t0

φpε (t, h∗ ) h∗p (t) dt =





(i, j )∈W p∈Pij

=



tf t0

=





μεij (h∗ )

(i, j )∈W

p∈Pij

(i, j )∈W

≤





μεij (h∗ )

(i, j )∈W p∈Pij

Therefore, .h∗ ∈ Λ1 is a solution of the DVI (4.70).

tf t0

h∗p (t) dt

μεij (h∗ ) · Qij

(i, j )∈W

=

μεij (h∗ ) h∗p (t) dt

p∈Pij tf t0

tf

hp (t) dt t0

φpε (t, h∗ ) hp (t) dt 

144

4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

4.5.2 With Endogenous Tolerances (VT-BR-DUE) We consider the more general case where the tolerances can depend on, in addition to O-D pairs, paths and established path departure rates. Mathematically, we define  .  p ε(·) = εij (·) : p ∈ Pij , (i, j ) ∈ W :

.

|P |

Λ1 → R+

 to the set of tolerances .ε (h), .∀p ∈ which maps a path departure rate vector .h ∈ Λ ij Pij , ∀(i, j ) ∈ W. We recall below the definition of BR-DUE with endogenous tolerances, also referred to as variable tolerance BR-DUE (VT-BR-DUE). p

Definition 4.2 (BR-DUE with Endogenous Tolerances) Given the vector of tol|P | erances .ε(·) : Λ1 → R+ , which is viewed as a mapping, a vector of departure ∗ rates .h ∈ Λ1 is a VT-BR-DUE associated with .ε(·) if, for all .(i, j ) ∈ W, we have h∗p (t)>0, p ∈ Pij ⇒ Ψp (t, h∗ )∈[vij (h∗ ), vij (h∗ )+εij (h∗ )] ∀ν (t) ∈ [t0 , tf ] (4.83) p

.

where .vij (h∗ ) is the essential infimum of the effective path delays between  origin-destination pair .(i, j ). We denote this equilibrium by VT-BR-DUE. Ψ, ε,  Λ1 , [t0 , tf ] . We define the mapping  |P | Φ ε : Λ1 → L2+ [t0 , tf ] ,

.

  h → Φpε (·, h) : p ∈ P

(4.84)

where " " # $ #% p p q Φpε (t, h) = max Ψp (t, h), vij (h) + εij (h) − εij (h) − min εij (h)

.

q∈Pij

(4.85) ∀p ∈ Pij , and .vij (h) denotes the essential infimum of the effective delay between O-D pair .(i, j ). We state and prove below the DVI equivalence theorem established for VT-BR-DUE.

.

|P |

Theorem 4.6 (DIV Formulation of VT-BR-DUE) Given .ε(·) : Λ1 → R+ , assume .Φpε (·, h), t ∈ [t0 , tf ] is measurable and strictly positive for all .p ∈ P and all .h ∈ Λ1 , where the operator .Φpε is defined in (4.84) and (4.85). A vector of departure rates (path flows) .h∗ ∈ Λ1 is a variable tolerance bounded rationality dynamic user equilibrium if and only if .h∗ solves the following differential varia-

4.5 DVI Formulation of DUE with Bounded Rationality

145

tional inequality .

find h∗ ∈ Λ1 such that

tf

Φpε (t, h∗ )(hp (t) − h∗p (t)) dt ≥ 0

t0

p∈P

∀ h ∈ Λ1

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

  DV I Φ ε , Λ1 , [t0 , tf ]

(4.86)

Proof (i) [Sufficiency] Given that .h∗ ∈ Λ1 is a solution of the differential variational inequality .DV I Φ ε , Λ1 , [t0 , tf ] , we have the following obvious relation:

tf

Φpε (t, h∗ )h∗p (t) dt ≤

.

p∈P

t0

p∈P

tf t0

Φpε (t, h∗ )hp (t) dt

for all .h ∈ Λ1 . In other words, .h∗ is the solution of the following optimal control problem: .

min J (h) =





(i, j )∈W p∈Pij

tf

t0



Φpε (t, h∗ ) hp (t) dt +

  μij Qij − yij (tf )

(i, j )∈W

(4.87) subject to .

d yij (t) = dt



hp (t)

∀(i, j ) ∈ W.

(4.88)

p∈Pij

yij (t0 ) = 0

∀(i, j ) ∈ W.

(4.89)

h ≥ 0

(4.90)

where each .μij is the dual variable for the terminal condition on .yij (·), .(i, j ) ∈ W. The Hamiltonian for problem (4.87)-(4.90) is H =





.

Φpε (t, h∗ ) hp +

(i, j )∈W p∈Pij



λij

(i, j )∈W



hp

p∈Pij

where the adjoint dynamics are: .

d ∂H λij (t) = − = 0 dt ∂yij

∀(i, j ) ∈ W,

t ∈ [t0 , tf ]

(4.91)

146

4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

The transversality conditions read λij (tf ) =

∂

.



(i, j )∈W

  μij Qij − yij (tf )

∂yij (tf )

= −μij

∀(i, j ) ∈ W,

t ∈ [t0 , tf ] (4.92)

Taken together, (4.91) and (4.92) imply that λij (t) ≡ − μij

∀(i, j ) ∈ W,

.

∀t ∈ [t0 , tf ]

(4.93)

According to the minimum principle, we have h∗ = arg min H

.

h

such that − h ≤ 0

for which the Kuhn-Tucker conditions are Φpε (t, h∗ ) + λij = ρp ≥ 0

∀p ∈ Pij ,

∀(i, j ) ∈ W,

∀ν (t) ∈ [t0 , tf ]. (4.94)

ρp (t) h∗p (t) = 0

∀p ∈ Pij ,

∀(i, j ) ∈ W,

∀ν (t) ∈ [t0 , tf ]. (4.95)

ρp (t) ≥ 0

∀p ∈ Pij ,

∀(i, j ) ∈ W,

∀ν (t) ∈ [t0 , tf ] (4.96)

.

where .ρp are dual variables for the non-negativity constraints. From (4.94) and (4.95), we have for all .(i, j ) ∈ W and .p ∈ Pij that h∗p (t) > 0, p ∈ Pij ⇒ Φpε (t, h∗ ) = − λij = μij

∀ν (t) ∈ [t0 , tf ]. (4.97)

Φpε (t, h∗ ) = ρp − λij = ρp + μij ≥ μij

∀ν (t) ∈ [t0 , tf ] (4.98)

.

from which we conclude that .∀(i, j ) ∈ W ! μij = essinf Φpε (t, h∗ ) : t ∈ [t0 , tf ], p ∈ Pij

.

= vij (h∗ ) + min {εij (h∗ )} q

q∈Pij

where the last equality was established in the proof of Theorem 3.3. In view of the definition of .Φpε (4.85), we restate (4.97) as h∗p (t) > 0, p ∈ Pij " # p p q ⇒ max Ψp (t, h∗ ), vij (h∗ ) + εij (h∗ ) − εij (h∗ ) + min {εij (h∗ )} .

q∈Pij

4.5 DVI Formulation of DUE with Bounded Rationality

147

= vij (h∗ ) + min {εij (h∗ )} q

q∈Pij

⇒ Ψp (t, h∗ ) ≤ vij (h∗ ) + εij (h∗ ) p

∀ν (t) ∈ [t0 , tf ],

∀(i, j ) ∈ W,

which is recognized as conditions describing a dynamic user equilibrium with variable tolerance bounded rationality, where .vij (h∗ ) is the essential infimum of effective path delays between origin-destination .(i, j ). (ii) [Necessity] For each origin-destination pair .(i, j ) ∈ W, define .μεij (h) to be the essential infimum of .Φpε (·, h) between .(i, j ). It is then easy to verify that .μεij (h) = q vij (h) + minq∈Pij {εij (h)} for all .(i, j ) ∈ W and for all .h ∈ Λ1 . For a VT-BR-DUE solution .h∗ we have for every .(i, j ) ∈ W that h∗p (t) > 0, p ∈ Pij ⇒ vij (h∗ ) ≤ Ψp (t, h∗ ) ≤ vij (h∗ ) + εij (h∗ ) p

.

which, in view of (4.85), translates to h∗p (t) > 0, p ∈ Pij ⇒ Φpε (t, h∗ ) = μεij (h∗ )

∀(i, j ) ∈ W

.

Therefore, given any .h ∈ Λ1 , we have



tf

.

(i, j )∈W p∈Pij

t0

Φpε (t,

h

∗

) h∗p (t) dt

=





(i, j )∈W p∈Pij

=



tf t0

=





μεij (h∗ )

(i, j )∈W

p∈Pij

(i, j )∈W

≤



tf t0

h∗p (t) dt

μεij (h∗ ) · Qij

(i, j )∈W

=

μεij (h∗ ) h∗p (t) dt



μεij (h∗ )

(i, j )∈W p∈Pij

p∈Pij tf t0

tf

hp (t) dt t0

Φpε (t, h∗ ) hp (t) dt (4.99)

Therefore, .h∗ ∈ Λ1 is a solution of the DVI (4.86).



148

4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

4.6 Fixed-Point Problem Formulation of DUE with Bounded Rationality 4.6.1 With Exogenous Tolerances (BR-DUE) We have established earlier that the BR-DUE problem with exogenous tolerance may be restated as the following equivalent DVI problem:

.

⎫ find h∗ ∈ Λ1 such that ⎪ ⎪

⎪ tf ⎬   ε ∗ ∗ φp (t, h )(hp − hp ) dt ≥ 0 DV I φ ε , Λ1 , [t0 , tf ] t ⎪ 0 ⎪ p∈P ⎪ ⎭ ∀ h ∈ Λ1

(4.100)

where the mapping .φ ε is defined as:  |P | φ ε : Λ1 → L2+ [t0 , tf ] ,

.

  h → φpε (·, h) : p ∈ P

(4.101)

and ! φpε (t, h) = max Ψp (t, h), vij (h) + εij

.

∀p ∈ Pij

∀(i, j ) ∈ W (4.102)

where .vij (h) denotes the essential infimum of the effective delay between O-D pair   |W | (i, j ) and .ε = εij : (i, j ) ∈ W ∈ R+ is the given vector of tolerances. The set of feasible path departure rate vectors .Λ1 is given in (4.66). It turns out that the BR-DUE problem with exogenous tolerances can be stated as an equivalent fixed-point problem in a functional space.

.

Theorem 4.7 (Fixed-Point Formulation of BR-DUE with Exogenous Tolerances) Assume that .φ ε (·, h) defined in (4.101)-(4.102) is measurable and strictly positive for all .p ∈ P and .h ∈ Λ1 . Then the fixed point problem   h∗ = PΛ1 h∗ − αφ ε (t, h∗ )

(4.103)

.

    is equivalent to .DV I φ ε , Λ1 , [t0 , tf ] and to BR-DUE. Ψ, ε, Λ1 , [t0 , tf ] , where .PΛ1 [·] is the minimum norm projection onto .Λ1 and .α > 0 is a fixed constant. Proof The fixed point problem (4.103) requires that 

 1 h∗ − αφ ε (t, h∗ ) − h2 : h ∈ Λ1 .h = arg min h 2 ∗



4.6 Fixed-Point Problem Formulation of DUE with Bounded Rationality

149

In other words, we seek a solution of the following optimal control problem

.

min J (h) = h

tf

t0

+

1 2

2  h∗ − αφpε (t, h∗ ) − h dt



(i, j )∈W p∈Pij



  μij Qij − yij (tf )

(4.104)

(i, j )∈W

subject to .

d yij = dt



hp (t)

∀(i, j ) ∈ W.

(4.105)

p∈Pij

yij (t0 ) = 0

∀(i, j ) ∈ W.

(4.106)

h ≥ 0

(4.107)

The Hamiltonian for the above optimal control problem is H =

.

1 2



2  h∗ − αφpε (t, h∗ ) − h +

(i, j )∈W p∈Pij

(i, j )∈W



λij

hp

(4.108)

p∈Pij

which is convex in its controls h and has no state dependence, so the minimum principle and supporting optimality conditions are both necessary and sufficient. The minimum principle for (4.108) has these Kuhn-Tucker conditions (the optimal solution is denoted with a double star): .

  h∗p − αφpε (t, h∗ ) − h∗∗ p · (−1) + λij = ρp

∀(i, j ) ∈ W,

p ∈ Pij . (4.109)

ρp h∗∗ p = 0

∀(i, j ) ∈ W,

p ∈ Pij . (4.110)

ρp ≥ 0

∀(i, j ) ∈ W,

p ∈ Pij (4.111)

By virtue of (4.103) and sufficiency of the minimum principle, we have that .h∗ ≡ h∗∗ . Therefore, we may restate (4.109) as αφpε (t, h∗ ) + λij = ρp

.

∀(i, j ) ∈ W,

p ∈ Pij

(4.112)

150

4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

Note that the adjoint equations and the associated transversality conditions are dλij ∂H = 0 ∀(i, j ) ∈ W. = (−1) dt ∂yij   ∂μij Qij − yij (tf ) = − μij λij (tf ) = ∂yij (tf )

(4.113)

.

∀(i, j ) ∈ W

(4.114)

Consequently, .λij (t) ≡ −μij , for all .(i, j ) ∈ W, .t ∈ [t0 , tf ]. Since .h∗ ≡ h∗∗ , we have by (4.110) that .h∗p > 0 implies .ρp = 0, .∀p ∈ P. Hence for almost every .t ∈ [t0 , tf ], we also have h∗p (t) > 0, p ∈ Pij ⇒ αφpε (t, h∗ ) = −λij = μij ⇒ φpε (t, h∗ ) =

.

μij α (4.115)

μ

Furthermore, (4.111) and (4.112) imply that .φpε (t, h∗ ) ≥ αij , .∀p ∈ Pij , .∀ν (t) ∈ μ [t0 , tf ]. This means that . αij is the essential infimum of .φ ε (h∗ ) within O-D pair .(i, j ) ∈ W: .

! μij = essinf φpε (t, h∗ ) : t ∈ [t0 , tf ], p ∈ Pij α

= vij (h∗ ) + εij

In view of the definition of .φpε (t, h∗ ), We equivalently state (4.115) as: μij h∗p (t) > 0, p ∈ Pij ⇒ φpε (t, h∗ ) = α ! ∗ ⇐⇒ max Ψp (t, h ), vij (h∗ ) + εij

.

= vij (h∗ ) + εij

⇐⇒ vij (h∗ ) ≤ Ψp (t, h∗ ) ≤ vij (h∗ ) + εij ∀ν (t) ∈ [t0 , tf ], which is recognized as the condition of BR-DUE.



4.6.2 With Endogenous Tolerances (VT-BR-DUE) We have established earlier that the BR-DUE problem with endogenous and variable tolerance (VT-BR-DUE) is equivalent to the following DVI problem:

.

⎫ find h∗ ∈ Λ1 such that ⎪ ⎪

⎪ tf ⎬   ε ∗ ∗ Φp (t, h )(hp − hp ) dt ≥ 0 DV I Φ ε , Λ1 , [t0 , tf ] t ⎪ ⎪ p∈P 0 ⎪ ⎭ ∀ h ∈ Λ1

(4.116)

4.6 Fixed-Point Problem Formulation of DUE with Bounded Rationality

151

where  |P | Φ ε : Λ1 → L2+ [t0 , tf ] ,

.

  h → Φpε (·, h) : p ∈ P

(4.117)

and ε .Φp (t,

" " # $ #% p p q h) = max Ψp (t, h), vij (h) + εij (h) − εij (h) − min εij (h) q∈Pij

(4.118) for all .p ∈ Pij , and .vij (h) denotes the essential infimum of the effective delay between O-D pair .(i, j ). The following theorem establishes the equivalence between the DVI and the FPP formulations of the BR-DUE problem with endogenous (variable) tolerances. Theorem 4.8 (Fixed-Point Formulation of VT-BR-DUE) Assume that .Φ ε (·, h) defined in (4.117)-(4.118) is measurable and strictly positive for all .p ∈ P and .h ∈ Λ1 . Then the fixed-point problem   h∗ = PΛ1 h∗ − αΦ ε (t, h∗ )

(4.119)

.

    is equivalent to .DV I Φ ε , Λ1 , [t0 , tf ] and to BR-DUE. Ψ, ε, Λ1 , [t0 , tf ] , where .PΛ1 [·] is the minimum norm projection onto .Λ1 and .α > 0 is a fixed constant. Proof The fixed-point problem (4.119) requires that h∗ = arg min



.

h

 1 h∗ − αΦ ε (·, h∗ ) − h2 : h ∈ Λ1 2



In other words, we seek a solution of the following optimal control problem

.

min J (h) = h

tf

t0

+

1 2



2  h∗ − αΦpε (t, h∗ ) − h dt

(i, j )∈W p∈Pij



  μij Qij − yij (tf )

(4.120)

(i, j )∈W

subject to .

d yij (t) = dt



hp (t)

∀(i, j ) ∈ W.

(4.121)

p∈Pij

yij (t0 ) = 0 h ≥ 0

∀(i, j ) ∈ W.

(4.122) (4.123)

152

4 The Differential Variational Inequality Formulation of Dynamic User Equilibria

The Hamiltonian for the above optimal control problem is 1 2

H =

.



2  h∗ (t) − αΦpε (t, h∗ ) − h(t) +

(i, j )∈W p∈Pij





λij (t)

(i, j )∈W

hp (t)

p∈Pij

(4.124) which is convex in its controls h and has no state dependence, so the minimum principle and supporting optimality conditions are both necessary and sufficient. The minimum principle for (4.124) has the following Kuhn-Tucker conditions .∀p ∈ Pij , ∀(i, j ) ∈ W: .

  − h∗p (t) − αΦpε (t, h∗ ) − h∗∗ (t) + λij (t) = ρp (t) p

∀ν (t) ∈ [t0 , tf ]. (4.125)

ρp (t) · h∗∗ p (t) = 0

∀ν (t) ∈ [t0 , tf ].

(4.126)

ρp (t) ≥ 0

∀ν (t) ∈ [t0 , tf ]

(4.127)

where the optimal solution that meets the K-T conditions is indicated with a doublestar. By virtue of (4.119) and sufficiency of the minimum principle, we have that ∗ ∗∗ .h ≡ h . Therefore, we may restate (4.125) as αΦpε (t, h∗ ) + λij (t) = ρp (t)

.

∀(i, j ) ∈ W,

p ∈ Pij

(4.128)

Note that the adjoint equations and the associated transversality conditions are .

dλij (t) ∂H = (−1) = 0 ∀(i, j ) ∈ W. dt ∂yij   ∂μij Qij − yij (tf ) = − μij λij (tf ) = ∂yij (tf )

(4.129) ∀(i, j ) ∈ W

(4.130)

Consequently, .λij (t) ≡ −μij , .∀(i, j ) ∈ W, .∀ν (t) ∈ [t0 , tf ]. Since .h∗ ≡ h∗∗ , we have by (4.126) that .h∗p (t) > 0 implies .ρp (t) = 0, .∀p ∈ P. Hence, for almost every .t ∈ [t0 , tf ], according to (4.128), we have h∗p (t) > 0, p ∈ Pij ⇒ αΦpε (t, h∗ ) = −λij = μij ⇒ Φpε (t, h∗ ) =

.

μij α (4.131)

μ

Furthermore, (4.127) and (4.128) imply that .Φpε (t, h∗ ) ≥ αij , .∀ν (t) ∈ [t0 , tf ]. This μ means that . αij is the essential infimum of .Φ ε (h∗ ) within O-D pair .(i, j ); that is, .

" # μij q = essinf Φpε (t, h∗ ) : t ∈ [t0 , tf ], p ∈ Pij = vij (h∗ ) + min {εij (h∗ )} α q∈Pij

References and Suggested Reading

153

where the last equality is established in the proof of Theorem 3.3. In view of the definition (4.118) of .Φpε (t, h∗ ), we equivalently write (4.131) as h∗p (t) > 0, p ∈ Pij

.

⇒ Φpε (t, h∗ ) = vij (h∗ ) + min {εij (h∗ )} p

p∈Pij

" # p p q ⇐⇒ max Ψp (t, h∗ ), vij (h∗ ) + εij (h∗ ) − εij (h∗ ) + min {εij (h∗ )} q∈Pij

= vij (h∗ ) + min {εij (h∗ )} q

q∈Pij

" # p p ⇐⇒ max Ψp (t, h∗ ), vij (h∗ ) + εij (h∗ ) = vij (h∗ ) + εij (h∗ ) ⇐⇒ vij (h∗ ) ≤ Ψp (t, h∗ ) ≤ vij (h∗ ) + εij (h∗ ) p

which is recognized as the condition of the VT-BR-DUE.

∀ν (t) ∈ [t0 , tf ] 

References and Suggested Reading Friesz, T. L. (2010). Dynamic optimization and differential games. New York: Springer. Friesz, T. L., Bernstein, D., Smith, T., Tobin, R., & Wie, B. (1993). A variational inequality formulation of the dynamic network user equilibrium problem. Operations Research, 41(1), 80–91. Friesz, T. L., Kim, T., Kwon, C., & Rigdon, M. A. (2011). Approximate network loading and dual-time-scale dynamic user equilibrium. Transportation Research Part B, 45(1), 176–207. Friesz, T. L., & Meiman, A. (2014). Dynamic user equilibria with elastic demand. Transportmetrica A: Transport Science, 10(7), 661–668. Luenberger, D. G. (1984). Linear and nonlinear programming. Reading, MA: Addison-Wesley. Mangasarian, O. (1969). Nonlinear programming. New York: McGraw-Hill. Pang, J. S., & Stewart, D. E. (2008). Differential variational inequalities. Mathematical Programming, Series A, 113(2), 345–424.

Chapter 5

Existence of Dynamic User Equilibria

This chapter presents an existence theory for a variety of dynamic user equilibria. Analytical DUE models tend to be of two varieties: (1) route choice (RC) user equilibrium (Friesz et al., 1989; Merchant and Nemhauser, 1978a,b; Mounce, 2006; Smith and Wisten, 1995; Zhu and Marcotte, 2000); and (2) simultaneous routeand-departure-time (SRDT) dynamic user equilibrium (Friesz et al., 1993, 2001, 2011, 2013; Ran et al., 1996; Wie et al., 2002). For both types of DUE models, the existence of a dynamic user equilibrium in continuous time remains a fundamental issue. A proof of DUE existence is a necessary foundation for qualitative analysis and computational studies. There are multiple means of expressing the Nash-like notion of a dynamic equilibrium in continuous-time, including the following 1. A variational inequality (Friesz et al., 1993; Smith and Wisten, 1994, 1995) 2. An equilibrium point of an evolution equation in an appropriate function space (Mounce, 2006; Smith and Wisten, 1995) 3. A nonlinear complementarity problem (Wie et al., 2002; Han et al., 2011) 4. A differential variational inequality (Friesz et al., 2001; Friesz and Mookherjee, 2006; Friesz et al., 2011, 2013) and 5. A differential complementarity system (Pang et al., 2011) The most obvious approach to establishing existence for any of the mathematical representations mentioned above is to convert the problem to an equivalent fixedpoint problem and then apply a version of Brouwer’s fixed-point existence theorem. Alternatively, one may use an existence theorem for the particular mathematical representation selected; it should be noted that most such theorems are derived by using Brouwer’s famous theorem. So, in effect, nearly all proofs of DUE existence employ Brouwer’s fixed-point theorem, either implicitly or explicitly. Approaches based on Brouwer’s theorem requires the set of feasible path flows (departure rates) under consideration to be compact and convex in a Banach space and typically involves an a priori bound on all the path flows. Proofs of existence that rely on © Springer Nature Switzerland AG 2022 T. L. Friesz, K. Han, Dynamic Network User Equilibrium, Complex Networks and Dynamic Systems 5, https://doi.org/10.1007/978-3-031-25564-9_5

155

156

5 Existence of Dynamic User Equilibria

a priori upper bounds of path flows are presented, for example, in Bressan and Han (2012b) and Zhu and Marcotte (2000). In this chapter, we present a general framework in which the existence of DUEs can be analyzed. In particular, we employ a much more general constraint relating path flows to a table of fixed trip volumes than has been previously considered when studying SRDT DUE. Moreover, in our study of existence, we do not invoke a priori bounds on the path flows to assure boundedness needed for application of Brouwer’s theorem. That is, a goal of our subsequent analysis is to investigate the existence of DUE without making the assumption of a priori bounds for departure rates. Note should be taken of the following fact: the boundedness assumption is uniquely relevant to DUE problems with simultaneous route and departure time choices; it is less of an issue for the route-choice DUE by virtue of problem formulation; that is, for RC DUE, the travel demand constraints are of the following form:  . hp (t) = Rij (t) ∀t ∈ [t0 , tf ], ∀(i, j ) ∈ W (5.1) p∈Pij

where .W is the set of origin-destination pairs, .Pij is the set of paths connecting (i, j ) ∈ W, and .hp (t) is the departure rate along path p. Furthermore, .Rij (t) represents the rate (not volume) at which travelers leave origin i with the intent of reaching destination j at time t; each such trip rate is assumed to be bounded from above. Since (5.1) is imposed pointwise and every path flow   .hp (·) is nonnegative, we are assured that each .h = hp : p ∈ Pij , (i, j ) ∈ W is automatically uniformly bounded. On the other hand, the SRCD user equilibrium imposes the following constraints on path flows:

.

 

tf

.

p∈Pij

hp (t) dt = Qij

∀(i, j ) ∈ W

(5.2)

t0

where .Qij ∈ R++ is the volume (not rate) of travelers departing node i with the intent of reaching node j . The integrals in (5.2) are interpreted as Lebesgue; hence, (5.2) alone is not enough to assure bounded path flows. This observation has been the major hurdle to proving existence without the a priori invocation of bounds on path flows. In our analytical framework, we will overcome this difficulty through careful analysis of the dynamic network loading sub-model and by investigating the effect of users’ disutility in shaping network flows, in a mathematically intuitive yet rigorous way.

5.1 Existence of Dynamic User Equilibrium with Fixed Demand

157

5.1 Existence of Dynamic User Equilibrium with Fixed Demand We begin our discussion with existence of simultaneous route-and-departure-time dynamic user equilibrium with fixed travel demand. In order to ensure the accuracy and rigor of our presentation, we begin with a brief review of some essential mathematical background, while referring the reader to Sect. 3.1 for more detailed explanations.

5.1.1 Mathematical Preparation The time horizon is a single period .[t0 , tf ] ⊂ R. A crucial component of the DUE model and its mathematical formulations is the path delay operator, which provides the path travel times given the path departure rates. We let Dp (t, h)

.

∀p ∈ P

be the time taken to traverse a link p, when the departure time is t, under the network load given by the vector of path departure rates h. .P is the set of paths employed by network users. Throughout this chapter, we stipulate that each path departure rate is a square-integrable function of the departure time; that is, hp (·) ∈ L2+ [t0 , tf ],

.

and

 |P | h(·) ∈ L2+ [t0 , tf ]

  where .h(·) = hp (·) : p ∈ P is the complete vector of path departure rates. The  |P | inner product of the Hilbert space . L2 [t0 , tf ] is defined as  .

u, v =

tf

(u(s))T v(s) ds =

t0

 p∈P

tf

up (s) · vp (s) ds

(5.3)

t0

where the superscript T denotes transpose of vectors. Moreover, the norm .

uL2 = u, u1/2

is induced by the inner product (5.3). In order to accommodate a more general notion of travel cost that will motivate on-time arrivals, we introduced the arrival penalty function .f (·) and the effective path delays .Ψp (t, h) defined as  . Ψp (t, h) = Dp (t, h) + f t + Dp (t, h) − TA

.

∀t ∈ [t0 , tf ],

p∈P

158

5 Existence of Dynamic User Equilibria

where .TA is the target arrival time. Note that, for convenience, .TA is assumed to be independent of destination. However, that assumption is easy to relax, and the consequent generalization of our model is a trivial extension. We interpret .Ψp (t, h) as the perceived travel cost of a driver departing at time t traveling along path p under the collective path departure rates h. Previously in Chaps. 3 and 4, we only |P | work with the assumption that .Ψ (·, h) : [t0 , tf ] → R++ is measurable and positive  2 |P | . These assumptions are natural for measure-theoretic for every .h ∈ L+ [t0 , tf ] arguments. In most of this chapter, we shall rely on one additional property of the effective delay operator, namely, continuity. The continuity of the effective delay operator is closely related to the dynamic network loading (DNL) sub-problem, which is related to the underlying traffic flow model and the network model. The continuity results for a variety of DNL models will be presented and shown later in Chap. 7. The existence of SRDT DUE is best analyzed when formulated as a variational inequality. That is, the SRDT DUE exists if and only if the following VI has a solution. ⎫ ∗ ∈ Λ such that ⎪ find h ⎪  ⎪ ⎬  

tf ∗ ∗ Ψ (t, h )(h − h )dt ≥ 0 V I Ψ, Λ, [t0 , tf ] . (5.4) p p p ⎪ p∈P t0 ⎪ ⎪ ⎭ ∀h ∈ Λ

Λ =

.

⎧ ⎨ ⎩



h ∈ L2+ ([t0 , tf ])

|P |

:

  p∈Pij

tf

hp (t) dt = Qij

t0

∀ (i, j ) ∈ W

⎫ ⎬ ⎭

(5.5) where .W is the set of origin-destination  pairs in the network. The variational inequality formulation .V I Ψ, Λ, [t0 , tf ] expressed above subsumes almost all DUE models regardless of the arc dynamic or the network loading models employed. The key foundation for our analysis of existence is the following theorem given in Browder (1968), which is recognized as an extension of the classical Brouwer’s fixed-point theorem to topological vector spaces. Theorem 5.1 Let K be a compact convex subset of the locally convex topological vector space E, T a continuous (single-valued) mapping of K into .E ∗ . Then there exists .u0 in K such that  .

for all .u ∈ K.

 T (u0 ), u − u0 ≥ 0

5.1 Existence of Dynamic User Equilibrium with Fixed Demand

159

Theorem 5.1 can be applied to show the existence of DUEs if (1) the effective path delay operator .Ψ can be shown to be continuous and (2) the feasible set .Λ is compact. Unfortunately, (2) is not true due to the infinite-dimensional nature of the underlying function space; the reader is referred to Example 2.5 in Sect. 2.1.3 for an explanation. To overcome this difficulty, we will employ a novel procedure based on successive finite-dimensional approximations of .Λ and rely on a topological argument to reach the desired existence result. Notice that such a challenge would not exist if the problem is conceived in a finite-dimensional space (i.e., discrete-time DUE) or only route choice is within the purview of travelers (route-choice DUE). The aforementioned difficulty is mainly due to the unboundedness of the path departure rates as we explained in our discussion of (5.2). In order to address this challenge, we employ a novel technique for re-expressing the effective path delays, which is detailed below.

5.1.2 Alternative Expression of the Effective Path Delay One of the main techniques we employ to tackle the existence of DUE is an alternative expression of the effective path delays .Ψp (t, h) for .t ∈ [t0 , tf ] and .p ∈ P, to be elaborated in this section. To fix the idea, let us recall the effective path delay:  . Ψp (t, h) = Dp (t, h) + f t + Dp (t, h) − TA

.

(5.6)

We rewrite (5.6) in a slightly different form. In particular, for each O-D pair .(i, j ) ∈ W, let us introduce the cost function .φij (·) : [t0 , tf ] → R+ , which is a function of departure time, and . ψij (·) : [t0 , tf ] → R+ , which is a function of arrival time. As we shall explain below, the users’ travel costs can be alternatively expressed using functions .φij (·) and .ψij (·). Given origin-destination pair .(i, j ) ∈ W, and any driver who departs from the origin at time .td and arrives at destination at .ta , his/her travel cost is expressed as .φij (td ) + ψij (ta ). We fix any vector of path departure rates .h ∈ Λ and introduce the path exit time function .τp (t) = t + Dp (t, h) where t denotes departure time. Then (5.6) can be equivalently written as     Ψp (t, h) = − t + τp (t) + f τp (t) − TA = φij (t) + ψij τp (t, h)

.

(5.7)

where . φij (t) = − t,

.

  . ψij (τp (t)) = τp (t) + f τp (t) − TA

(5.8)

Remark 5.1 In Bressan and Han (2011, 2012a,b), the travel costs are measured in terms of .φij (·) and .ψij (·). In other words, the travel cost (5.6) can be alternatively evaluated as a sum of costs at the beginning and end of journey of each driver.

160

5 Existence of Dynamic User Equilibria

We consider a general road network expressed as a graph .G(V, A), where .V and A denote the sets of nodes and links, respectively. For each O-D pair .(i, j ) ∈ W, we make the following two assumptions on .φij (·) and .ψij (·), as well as the underlying traffic flow model.

.

A1.

For each .(i, j ) ∈ W, .φij (·) and .ψij (·) are continuous on .[t0 , tf ]; .φij (·) is monotonically decreasing; and .ψij (·) is monotonically increasing. In addition, we assume that .φij (·) is Lipschitz continuous with constant .Lij ; and there exists .Δij > 0 such that ψij (t2 ) − ψij (t1 ) ≥ Δij (t2 − t1 )

.

A2.

∀ t0 ≤ t1 < t2 ≤ tf

(5.9)

Each link .a ∈ A in the network has a finite flow capacity .Ma < ∞.

Inequality (5.9) requires that the arrival cost function .ψij (·) is strictly increasing, and the rate of increase is bounded below by .Δij . If .ψij (·) is continuously d differentiable, this is equivalent to requiring that . dt ψij (t) ≥ Δij > 0 for .t ∈ d [t0 , tf ], which is further equivalent to . dt ψij (t) > 0 t ∈ [t0 , tf ], due to the compactness of the interval .[t0 , tf ]. As a special case, given the effective delay of the form (5.7) and (5.8), A1 reduces to the following assumption: A1’. .f (·) is continuous on .[t0 , tf ] and satisfies f (t2 ) − f (t1 ) ≥ Δ(t2 − t1 )

.

∀t0 ≤ t1 < t2 ≤ tf

for some .Δ > −1. Assumption A2 applies to a wide range of traffic flow models that assume a finite flow capacity for each link, e.g., the Vickrey model (Vickrey, 1969), the LWR-Lax model (Friesz et al., 2013), and the Lighthill-Whitham-Richards model (Lighthill and Whitham, 1996; Richards, 1956; Han et al., 2012). Notice that such an assumption does not hold for the link delay model proposed in Friesz et al. (1993) Remark 5.2 The proposed alternative formulation of path travel cost subsumes another well-known class of travel cost functions that are employed, for example, in Pang et al. (2011) and Yao et al. (2010). Namely, given positive constants .α, β, and .γ , the path travel cost of road users is expressed as  α(ta − td ) +

.

β(T − ta )

ta ≤ TA

γ (ta − T )

ta > TA

where .td and .ta denote departure time and arrival time, respectively and .TA is the desired arrival time. It is assumed that .γ > α > β. This definition of travel cost can

5.1 Existence of Dynamic User Equilibrium with Fixed Demand

be rewritten as  α(ta − td ) +

.

β(TA − ta )

t a ≤ TA

γ (ta − TA )

t a > TA

 = − α td +

161

(α − β)ta + βTA

t a ≤ TA

(α + γ )ta − γ TA

t a > TA

= φ(td ) + ψ(ta ) where  . .φ(td ) = − α td ,

. ψ(ta ) =

(α − β)ta + βTA

t a ≤ TA

(α + γ )ta − γ TA

t a > TA

One can easily check that .φ(·) and .ψ(·) defined in such a way satisfy assumption A1. In view of the preceding assumptions A1 and A2, we are prompted to define the following constants for a given network. .

= φmax

.

.

= ψmin

.

max Lij > 0

(5.10)

min Δij > 0

(5.11)

(i, j )∈W

(i, j )∈W

. M max = max Ma < + ∞

.

a∈A

(5.12)

5.1.3 The Existence Proof The result described as Browder’s existence theorem (Theorem 5.1) will be the key ingredient of the existence proof of the DUE solution. In order to apply such a theorem, a notion of the continuity of the delay operator is required. We invoke the following definition of continuity, which is tailored to suit the topological argument that we will employ for proving the existence. A3.

For any sequence of departure rate vectors .{h(n) }n≥1 ⊂ Λ that are uniformly bounded by a positive constant and converge weakly to .h∗ ∈ Λ, the corresponding effective path delays .Ψp (t, hn ) converge to .Ψp (t, h∗ ) uniformly for all .p ∈ P and .t ∈ [t0 , tf ] as .n → +∞.

Assumption A3 is one type of continuity that is suitable for the topological argument we will adopt to show the existence of a variety of DUE models. It is closely related to the network performance model, i.e., the dynamic network loading sub-problem

162

5 Existence of Dynamic User Equilibria

of DUE. Later in this book, we will show that this assumption holds when a number of traffic flow models are considered. But for now, we will invoke this assumption without proof. In order to apply Theorem 5.1, we recap several key results from functional analysis that facilitate our presentation. In particular, we note the following facts without proofs. The reader is referred to Royden and Fitzpatrick (1988) for more detailed discussion on these subjects. Proposition 5.1 The space of square-integrable real-valued functions on a compact interval .[t0 , tf ], denoted by .L2 [t0 , tf ], is a locally convex topological vector |P |  space. In addition, the .|P|-fold product of these spaces, denoted by . L2 [t0 , tf ] , is also a locally convex topological vector space. Finally, the product space  2 |P | . L [t0 , tf ] × R|W | is again a locally convex topological vector space.  |P | Proposition 5.2 The dual space of . L2 [t0 , tf ] has a natural isomorphism with  2 |P | . L [t0 , tf ] . The dual space of the Euclidean space .R|W | consisting of columns of .|W| real numbers is interpreted as the space consisting of rows of .|W| real  |P | × R|W | is again numbers. As a consequence, the dual space of . L2 [t0 , tf ]  2 |P | . L [t0 , tf ] × R|W | . Proposition 5.3 In a metric space (therefore topological vector space), the notion of compactness is equivalent to the notion of sequential compactness; that is, every infinite sequence has a convergent subsequence. Therefore, the underlying topological vector space E from Theorem 5.1 will be  |P | instantiated by . L2 ([t0 , tf ]) , which is a locally convex topological vector space.  |P | ∗ The dual space .E will again be . L2 ([t0 , tf ]) . Theorem 5.2 (Existence of DUE with fixed demand) Let assumptions A1, A2, and A3 hold. Then the dynamic user equilibrium problem with fixed demand (Definition 3.1) has a solution. Proof The proof is divided into four parts. Part 1.

Our strategy for demonstrating existence is to adapt Theorem 5.1 to the  |P | locally convex topological vector space . L2 ([t0 , tf ]) and its subset .Λ. However, .Λ is bounded, closed, and convex, but not compact in  2 |P | . L ([t0 , tf ]) . To overcome this difficulty, we invoke the following finite-dimensional approximation procedure.

5.1 Existence of Dynamic User Equilibrium with Fixed Demand

163

We consider for each .n ≥ 1 the uniform partition of interval .[t0 , tf ] into n sub-intervals1

Part 2.

t0 = t 0 < t 1 < t 2 . . . < t n = tf

.

t i − t i−1 =

t f − t0 n

i = 1, . . . , n

Then, we consider the following sequence of finite-dimensional subsets . Λn =

.

 h∈Λ:

hp (·) is constant on [t i−1 , t i )

∀p ∈ P



⊂ Λ (5.13)

We claim that for each .n ≥ 1, .Λn is compact and convex in  2 |P | . L ([t0 , tf ]) . Indeed, given any .hn,1 , hn,2 ∈ Λn , and .α ∈ [0, 1], n,1 .α h + (1 − α) hn,2 is clearly nonnegative and constant on each .[t i−1 , t i ] for . i = 1, . . . , n. In addition, for any origin-destination pair .(i, j ) ∈ W, by definition (5.5), we have  

tf

.

p∈Pij

α hn,1 (t) + (1 − α) hn,2 (t) dt = α Qij + (1 − α) Qij = Qij

t0

This verifies that .Λn is convex. To see its compactness, we define the map n×|P | .μ : Λn → R+ , h → (ai, p : 1 ≤ i ≤ n, p ∈ P) where each vector i n , .(a1,p , . . . , an,p ) is the coordinate of .hp (·) under the natural basis .{en } i=1 where  1 t ∈ [t i−1 , t i ) i .en (t) = 0 else Clearly, the map .μ is one-to-one. We now consider any sequence  {hn,ν }ν≥1 ⊂ Λn , and the sequence of their images .{μ(hn,ν ) ν≥1 ⊂   n×|P | . By (5.5), . μ(hn,ν ) ν≥1 is uniformly bounded by the following R+ quantity

.

.

n Qij (i,j )∈W tf − t0 max

(5.14)

By the Bolzano-Weierstrass theorem, there exists a convergent subse  n,ν quence . μ(h ) ν ≥1 . By construction, the corresponding subsequence 1 The choice of the number of sub-intervals is quite flexible, as long as the size of the sub-intervals approaches zero as .n → +∞.

164

5 Existence of Dynamic User Equilibria



 hn,ν ν ≥1 must converge uniformly to some .hˆ n . In view of the compact interval .[t0 , tf ], we conclude that this convergence is also in the . · L2 norm. It now remains to show .hˆ n ∈ Λn ; then our claim follows from sequential compactness of .Λn (see Sect. 2.1.3). Clearly .hˆ n ≥ 0 and is constant on the sub-intervals .[t i−1 , t i ), i = 1, . . . , n. Moreover, since n(ν ) are uniformly bounded, by the dominated convergence theorem, we .h have   tf   tf n,ν .Qij = lim hp (t) dt = hˆ np (t) dt

.

ν →∞

p∈Pij

t0

p∈Pij

t0

This implies .hˆ n ∈ Λn . Thereby, the claim that each .Λn is compact is substantiated. Part 3. Before Theorem 5.1 can be applied, we need to show that the operator .Ψ is continuous on .Λn . Fix .n ≥ 1 and consider any converging sequence n,ν } ˆ n ∈ Λn . Each of the entire sequence .{h ν≥1 ⊂ Λn with the limit .h n,ν of functions .{h }ν≥1 is uniformly bounded by the constant in (5.14). Moreover, their strong convergence implies weak convergence to .hˆ n . Thus, we invoke assumption A3 and conclude that the effective path delays n,ν ) → Ψ (t, h ˆ n ) uniformly for all .t ∈ [t0 , tf ] and .p ∈ P. The .Ψp (t, h p fact that .[t0 , tf ] is compact implies that the convergence holds also in the 2 .L sense; i.e., Ψ (hn,ν ) − Ψ (hˆ n )L2 → 0

as

.

Part 4.

ν → +∞

This shows the strong continuity of the operator .Ψ on .Λn . For each .n ≥ 1, we can now apply Theorem 5.1 to .Λn to obtain .hn,∗ ∈ Λn such that  .

Ψ (hn,∗ ), hn − hn,∗



≥ 0

∀ hn ∈ Λn

(5.15)

where . ,  is the inner product defined in (5.3). It is easy to observe that j j +1 ], (5.15) implies that, given any .p ∈ P, if .hn,∗ p (t) > 0 for .t ∈ [t , t then 

t j +1

.

tj

 Ψp (t, hn,∗ ) dt =

min

t k+1

0≤k≤n t k

Ψp (t, hn,∗ ) dt

(5.16)

In view of the definitions (5.10)–(5.12), we choose any constant .M such that M >

.

3M max φmax

ψmin

(5.17)

5.1 Existence of Dynamic User Equilibrium with Fixed Demand

165

We then claim that, for all .n ≥ 1, there must hold .hn,∗ p (t) ≤ M for all .t ∈ [t0 , tf ], p ∈ P. Otherwise, by contradiction, we assume that there exists some .ν ≥ 1 and some .0 ≤ i ≤ ν such that, for some path .p ∈ P and time interval .[t i , t i+1 ], we have hv,∗ (t) ≡ η > M

.

t ∈ [t i , t i+1 ]

By choosing .[t0 , tf ] large enough, we can assume that .i ≥ 1. We now consider the interval .[t i−1 , t i ], and the quantity . sup Ψp (t, hv,∗ ). By t∈[t i−1 ,t i ]

possibly modifying the value of the function .Ψp (·, hv,∗ ) at one point without affecting the integrations in (5.16), we can obtain .t ∗ ∈ [t i−1 , t i ] such that Ψp (t ∗ , hv,∗ ) =

sup

.

t∈[t i−1 ,t i ]

Ψp (t, hv,∗ )

Now we let .τp (t, hv,∗ ) = t + Dp (t, hv,∗ ) be the arrival time associated with path p and departure time t. According to the first-in-first-out (FIFO) principle and assumption A2, we deduce that for any .t ∈ [t i , t i+1 ], τp (t, hv,∗ ) − τp (t ∗ , hv,∗ ) ≥

.

(t − t i ) η M max

(5.18)

This, together with (5.11), imply that     ψij τp (t, hv,∗ ) − ψij τp (t ∗ , hv,∗ )

.

  (t − t i ) η

≥ ψmin τp (t, hv,∗ ) − τp (t ∗ , hv,∗ ) ≥ ψmin · M max

(5.19)

Inequality (5.19) implies Ψp (t, hv,∗ ) − Ψp (t ∗ , hv,∗ )

.

≥

ψmin η

(t − t i ) − φmax (t − t ∗ ) max M

∀ t ∈ [t i , t i+1 ]

(5.20)

Integrating (5.20) with respect to t over interval .[t i , t i+1 ] and a simple calculation yield 

t i+1

.

ti

Ψp (t, hv,∗ ) dt − (t i+1 − t i )Ψp (t ∗ , hv,∗ )

 

η (t i+1 − t i )2 ψmin t i + t i+1 i+1 i ∗ · max + (t ≥ − t )φmax · t − 2 M 2 (5.21)

166

5 Existence of Dynamic User Equilibria

Since .t ∗ ∈ [t i−1 , t i ], we have .t ∗ − (t i + t i+1 )/2 ≥ −3/2(t i+1 − t i ). Inequality (5.21) then implies 

t i+1

.

ti

Ψp (t, hv,∗ ) dt − (t i+1 − t i )Ψp (t ∗ , hv,∗ ) (t i+1 − t i )2 ≥ 2



ψmin η

− 3φmax max M

 > 0

(5.22)

This yields the following contradiction to (5.16) and hence (5.15) holds: 



t i+1

Ψp (t, h

.

ti

v,∗

ti

) dt > t i−1

Ψp (t, hv,∗ ) dt

This substantiates our claim that the path departure rates .hn,∗ p (t) are uniformly bounded for all t, p, and n. Part 5. With the point-wise uniform bound .M on the path departure rate vectors n,∗ for all .n ≥ 1, by taking a subsequence if necessary, one can assume .h  |P | the weak convergence .hn,∗ → h∗ ∈ L2+ [t0 , tf ] . According to assumption A3, we have that .Ψp (t, hn,∗ ) → Ψp (t, h∗ ) uniformly for all n,∗ ) → Ψ (h∗ ) strongly. .t ∈ [t0 , tf ] and .p ∈ P, and thus .Ψ (h Finally, we show that  .

Ψ (h∗ ), h − h∗



≥ 0

∀h ∈ Λ

Indeed, we consider any .h ∈ Λ and choose a sequence .hn ∈ Λn such that .hn − hL2 → 0 as .n → ∞. For every n, by (5.15), one has  .

Ψ (hn,∗ ), hn − hn,∗



≥ 0

(5.23)

For any n, 

   Ψ (h∗ ), h − Ψ (h∗ ), h∗       = Ψ (h∗ ), h − Ψ (hn,∗ ), hn + Ψ (hn,∗ ), hn − hn,∗     + Ψ (hn,∗ ) − Ψ (h∗ ), hn,∗ + Ψ (h∗ ), hn,∗ − h∗       ≥ Ψ (h∗ ), h − Ψ (hn,∗ ), hn + Ψ (hn,∗ ) − Ψ (h∗ ), hn,∗   + Ψ (h∗ ), hn,∗ − h∗ .

(5.24)

5.2 Existence of Dynamic User Equilibrium with Elastic Demand

167

By strong convergence .Ψ (hn,∗ ) → Ψ (h∗ ) and .hn → h, the first term in the square brackets converges to zero. The second term of (5.24) converges to zero due to  .

Ψ (hn,∗ ) − Ψ (h∗ ), hn,∗



≤ Ψ (hn,∗ ) − Ψ (h∗ )L2 · hn,∗ L2

and the fact that the first factor converges to zero by strong convergence, and the second factor is uniformly bounded for all n. Finally, the third term in (5.24) converges to zero by weak convergence. Thus, taking the limit .n → ∞ in (5.24) yields    Ψ (h∗ ), h − Ψ (h∗ ), h∗ ≥ 0

 .

Since h is arbitrary, .h∗ is a solution of the variational inequality.



5.2 Existence of Dynamic User Equilibrium with Elastic Demand The existence theory for DUE with elastic demand (hereafter referred to as E-DUE) is again facilitated by the variational inequality formulation presented in Sect. 3.3.2:  .

p∈P

tf

t0

⎫  such that find (h∗ , Q∗ ) ∈ Λ ⎪ ⎪ ⎪    ⎬  ∗ ∗ ∗ ∗ Ψp (t, h )(hp − hp )dt − Θij Q Qij − Qij ≥ 0 ⎪ ⎪ (i, j )∈W ⎪ ⎭  ∀(h, Q) ∈ Λ (5.25)

As our first step toward the formal proof of existence, we need to rewrite (5.25) as a generic variational inequality in terms of an inner product in a Hilbert space, i.e., in the form .

T (u0 ), u − u0  ≥ 0

before Theorem 5.1 can be applied. This will be done in Sect. 5.2.1 below.

5.2.1 The Variational Inequality Formulation in an Extended Hilbert Space In view of (5.25), we are prompted to define the product space  |P | . E = L2 [t0 , tf ] × R|W | ,

.

168

5 Existence of Dynamic User Equilibria

which is a space with the inherited inner product defined as follows.

.

|P |  .  X, Y E = i=1

tf

ξi (t) · ηi (t) dt +

t0

|W | 

(5.26)

uj vj

j =1

 ξ1 (·), . . . , ξ|P | (·), u1 , . . . , u|W | ∈ E   Y = η1 (·), . . . , η|P | (·), v1 , . . . , v|W | ∈ E

X =



Accordingly, we define the norm on this space to be: . 1/2 XE = X, XE

.

In the following proposition, we show that the inner product .·, ·E and the norm  · E are well defined, and the resulting space E is indeed a Hilbert space.

.

Proposition 5.4 The inner product .·, ·E and norm . · E are well defined. In addition, the space E, equipped with .·, ·E and the induced metric, is a Hilbert space over .R, the set of real numbers. Proof A well-defined inner product over .R must satisfy, for all .X, Y, Z ∈ E, the following: 1. Symmetry, i.e., .X, Y E = Y, XE ; 2. Linearity, i.e., .aX, Y E = a X, Y E for all .a ∈ R, and .X + Y, ZE = X, Z + Y, ZE ; 3. Positive-definiteness, i.e., .X, XE ≥ 0 and .X, XE = 0 ⇒ X = 0. It is straightforward to verify that the inner product defined in (4.47) satisfies all these three conditions and, thus, is well defined. Consequently, the induced norm  2 |P | . · E is also well defined. Finally, since both . L [t0 , tf ] and .R|W | are complete metric spaces, their product space E is also a complete metric space and hence a Hilbert space.  We introduce the following feasible set which is a subset of the extended Hilbert space E:    |W | 2 |P |  .Λ = (h, Q) ∈ (L+ [t0 , tf ]) × R+ : ∀(i, j ) ∈ W



p∈Pij

tf

hp (t) dt = Qij

t0

Recalling from Sect. 3.3.1, the inverse demand function .Θ = (Θij : (i, j ) ∈ W), we define . Θ − = (−Θij : (i, j ) ∈ W) :

.

|W |

|W |

R++ −→ R−−

5.2 Existence of Dynamic User Equilibrium with Elastic Demand

169

 −→ E via We define the mapping .F : Λ X = (h, Q) →

.

  Ψ (·, h), Θ − (Q)

   .Ψ (·, h) ∈ L2+ [t0 , tf ] |W | , and Θ − (Q) ∈ R|−W | . Such a where .(h, Q) ∈ Λ, mapping is clearly well defined. With the preceding discussion, the DVI formulation of the DUE problem with elastic demand is readily rewritten as the following infinite-dimensional variational inequality in extended Hilbert space:  such that find X∗ ∈ Λ .

F(X∗ ), X − X∗ E ≥ 0  ∀X ∈ Λ

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

   t 0 , tf V I F, Θ, Λ,

(5.27)

Problem (5.27) is written as a generic VI, which allows us to establish existence using Browder’s fixed-point theorem.

5.2.2 The Existence Proof The assumptions for the existence of E-DUE are the same as the ones for the existence of DUE with fixed demand. These assumptions are recapped below. A1.

For each .(i, j ) ∈ W, .φij (·) and .ψij (·) are continuous on .[t0 , tf ]; .φij (·) is monotonically decreasing; and .ψij (·) is monotonically increasing. In addition, we assume that .φij (·) is Lipschitz continuous with constant .Lij and there exists .Δij > 0 such that ψij (t2 ) − ψij (t1 ) ≥ Δij (t2 − t1 )

.

A2. A3.

∀ t0 ≤ t1 < t2 ≤ tf

(5.28)

Each link .a ∈ A in the network has a finite flow capacity .Ma < ∞. For any sequence of departure rate vectors .{h(n) }n≥1 ⊂ Λ that are uniformly bounded by a positive constant and converge weakly to .h∗ ∈ Λ, the corresponding effective path delays .Ψp (t, hn ) converge to .Ψp (t, h∗ ) uniformly for all .p ∈ P and .t ∈ [t0 , tf ] as .n → +∞.

When we apply Browder’s fixed-point theorem to the VI (5.27) in search for a solution, we employ an approach similar to the fixed-demand case by invoking  Details are provided in the existence finite-dimensional approximations of the set .Λ. theorem proof below. Theorem 5.3 (Existence of E-DUE) Let assumptions (A1)–(A3) hold. If, in addi|W | |W | tion, the inverse demand function .Θ : R+ → R++ is continuous, then the E-DUE problem has a solution.

170

5 Existence of Dynamic User Equilibria

Proof The proof is completed in several steps. Step 1.

Since Theorem 5.1 cannot be directly applied to obtain a solution of the infinite-dimensional VI, let us instead consider finite-dimensional  More specifically, consider, for each .n ≥ 1, the approximations of .Λ. uniform partition of .[t0 , tf ] by n sub-intervals .I 1 , . . . , I n . Define the  finite-dimensional subset of .Λ:  .  n = : Λ h1 (·), . . . , h|P | (·), Q1 , . . . , Q|W | ∈ Λ

.

! hi (·) is constant on each I j

∀1 ≤ j ≤ n,

∀1 ≤ i ≤ |P| (5.29)

n ⊂ Λ,  thus all the assumptions regarding .F or .Ψ continue to Clearly .Λ hold on this smaller set. It is not restrictive to assume that there is an upper bound on the elastic demand for each origin-destination pair. That is, there exists a vector .U =   |W | Uij ∈ R++ such that 0 ≤ Qij ≤ Uij

.

∀(i, j ) ∈ W,

 ∀(h, Q) ∈ Λ

n defined as such is convex and compact in .Λ  for each We claim that .Λ " " " .n ≥ 1. We begin with verifying convexity. Let .X = (h, Q) and .X = n . Given any .α ∈ (0, 1), we have that (h, Q) be any two elements of .Λ  

tf

.

p∈Pij

  α" hp (t) dt + (1 − α) hp (t) dt

t0

"ij + (1 − α) Qij = αQ

∀(i, j ) ∈ W

Moreover, .α  hp (·) + (1 − α) hp (·) clearly remains constant on each sub"+ interval .I j , j = 1, . . . , n, for all .p ∈ P. We thus conclude that .α X  (1 − α)X ∈ Λn . Next, let us investigate compactness. From now on let us fix .n ≥ 1. In light of Proposition 5.3, it suffices to establish sequential compactness   n . We consider an arbitrary infinite sequence . Xk n where ⊂Λ for .Λ k≥1   k ) ∈ Rn be k k k k .X = h , Q . For each .k ≥ 1 and .p ∈ P, let .μp = (μ + p,j such that μkp,j = hkp (t)

.

t ∈ Ij,

∀j = 1, . . . , n

5.2 Existence of Dynamic User Equilibrium with Elastic Demand

171

n|P |

We then define .μk ∈ R+ to be the concatenation of all vectors .μkp , p ∈ P. We also notice that the vectors .μk , k ≥ 1 are uniformly bounded by the constant .

max Uij ·

(i, j )∈W

n tf − t0

Thus by the Bolzano-Weierstrass theorem, there exists a convergent

 subsequence . μk k ≥1 . It is immediately verifiable that the corresponding

Step 2.

subsequence .hk converges uniformly on .[t0 , tf ] and also in the .L2 norm. Moreover, by virtue of the uniform bounds .Uij , (i, j ) ∈ W, there    

exists a further subsequence . k

⊂ k such that .Qk is a convergent subsequence according to the Bolzano-Weierstrass theorem. Thus, the

 subsequence . Xk k

≥1 converges with respect to the norm induced by inner product (4.47). n , due to the upper bound For each .n ≥ 1, and any .Xn = (hn , Qn ) ∈ Λ .Uij on the elastic demand .Qij for all .(i, j ) ∈ W, the departure rate vector .hn is uniformly bounded. Thus, (A3) implies that .Ψ (·, hn ) → Ψ (·, h∗ ) in the .L2 norm as .n → ∞. Combining this with the fact that .Θ n . is continuous, we conclude that .F defined in (4.50) is continuous on .Λ n,∗ n,∗ n,∗ Thus, Theorem 5.1 asserts that there exists some .X = (h , Q ) ∈ n such that Λ .

  n,∗   F X , Xn − Xn,∗ E ≥ 0

n ∀Xn ∈ Λ

(5.30)

As a consequence of (5.30), we have 

tf

.

p∈P

≤

t0



p∈P

tf t0

  Ψp t, hn,∗ hn,∗ p (t) dt −



Θij [Qn,∗ ]Qn,∗ ij

(i, j )∈W

  Ψp t, hn,∗ hnp (t) dt −



Θij [Qn,∗ ]Qnij

(5.31)

(i, j )∈W

    n . In particular, for all .hn ∈ L2+ [t0 , tf ] |P | such for all . hn , Qn ∈ Λ that   tf . hnp (t) dt = Qn,∗ ∀(i, j ) ∈ W, ij p∈Pij

t0

172

5 Existence of Dynamic User Equilibria

inequality (5.31) becomes 

   Ψp t, hn,∗ hn,∗ (t) dt ≤ p

tf

.

p∈P

t0

tf

t0

p∈P

  Ψp t, hn,∗ hnp (t) dt (5.32)

Recall that .hn,∗ (·) is piecewise constant, thus (5.32) implies the following:  n,∗ .hp (t)

t ∈I

> 0,

k

⇒

 Ik

Ψp (t, h

n,∗

) dt =

min

1≤j ≤n I j

Ψp (t, hn,∗ ) dt (5.33)

Step 3.

Step 4.

for all .p ∈ Pij , .(i, j ) ∈ W. We invoke assumptions A1 and A2 and show that there exists a uniform upper bound .M on .hn,∗ p (·) for all .n ≥ 1 and .p ∈ P. The proof is almost identical to that in Part 4 of the proof of Theorem 5.2 and thus is not repeated here. With the point-wise uniform upper bound .M on the path departure rate vectors .hn,∗ for all .n ≥ 1, by taking a subsequence if necessary, one can  |P | assume weak convergence .hn,∗ → h∗ ∈ L2+ [t0 , tf ] . Let ∗ .Qij

. =

 

tf

t0

p∈Pij

h∗p (t) dt

 .  Q∗ = Q∗ij : (i, j ) ∈ W

∀(i, j ) ∈ W,

We have that for any .(i, j ) ∈ W, Q∗ij =

 

tf

.

t0

p∈Pij

h∗p (t) dt =

 

lim

n→∞

p∈Pij

tf t0

hn,∗ p (t) dt =

lim Qn,∗ ij

n→∞

. . Therefore, we have that .Xn,∗ = (hn,∗ , Qn,∗ ) → X∗ = (h∗ , Q∗ ) weakly. n,∗ According to (A3), we have that .Ψp (t, h ) → Ψp (t, h∗ ) uniformly for all .t ∈ [t0 , tf ] and .p ∈ P, and thus .F(Xn,∗ ) → F(X∗ ) strongly. Finally, we show that  .

F(X∗ ), X − X∗

 E

≥ 0

 ∀X ∈ Λ

 and choose a sequence .Xn ∈ Λ n such that .Xn − Indeed, we consider any .X ∈ Λ, XE → 0 as .n → ∞. For every n, by (5.30), one has .

  n,∗   F X , Xn − Xn,∗ E ≥ 0

(5.34)

5.3 An Example of Non-existence of DUE

173

This leads to the following for all n: 

   F(X∗ ), X E − F(X∗ ), X∗ E        = F(X∗ ), X E − F(Xn,∗ ), Xn E + F(Xn,∗ ), Xn − Xn,∗ E     + F(Xn,∗ ) − F(X∗ ), Xn,∗ E + F(X∗ ), Xn,∗ − X∗ E     ≥ F(X∗ ), X E − F(Xn,∗ ), Xn E . (5.35)   n,∗ ∗ n,∗ + F(X ) − F(X ), X E . (5.36)   ∗ n,∗ ∗ + F(X ), X − X E (5.37) .

By strong convergence .F(Xn,∗ ) → F(X∗ ) and .Xn → X, (5.35) converges to zero. The term (5.36) converges to zero due to  .

F(Xn,∗ ) − F(X∗ ), Xn,∗

 E

# # # # ≤ #F(Xn,∗ ) − F(X∗ )#E · #Xn,∗ #E

and the fact that the first factor converges to zero by strong convergence, and the second factor is uniformly bounded for all n. Finally, (5.37) converges to zero by weak convergence. Thus, taking the limit .n → ∞ in (5.35)–(5.37) yields  .

   F(X∗ ), X E − F(X∗ ), X∗ E ≥ 0

Since X is arbitrary, .X∗ is a solution of the variational inequality. 

5.3 An Example of Non-existence of DUE In this section, we provide a concrete example where a fixed-demand DUE solution, as we presented in Definition 3.1, does not exist. This example is given by Bressan and Han (2011). We consider a network consisting of one single link and one O-D pair. The link flow dynamics are governed by the Lighthill-Whitham-Richards model (Lighthill and Whitham, 1996; Richards, 1956) with a Greenshields (quadratic) fundamental diagram:  .f (ρ) = ρ 1−

ρ ρj am

 v0

ρ ∈ [0, ρj am ]

where .ρ denotes vehicle density and .f (ρ) is called the fundamental diagram, which describes the relationship between aggregate vehicle flow and density. .ρj am is the

174

5 Existence of Dynamic User Equilibria

. ρ v0 jam density; and .v0 denotes the free-flow speed. The quantity .C = j am represents 4 the maximum flow allowed and is referred to as the flow capacity. The path departure rate is denoted by .h(t) where the subscript p is omitted since there is only one path. In case .h(t) > C, a Vickrey-type point queue is assumed to be present. The queuing dynamic is defined as:  .

q(t) ˙ = h(t) − g(t),

g(t) =

min{h(t), C}

if q(t) = 0

C

if q(t) > 0

where .g(t) denotes the rate at which cars are discharged from the queue, if any. The first identity represents the conservation of flow. According to the second identity, if the queue is empty, then the discharge flow is equal to the departure rate as constrained by the flow capacity; if the queue is non-empty, then cars are discharged at the maximum rate possible. We refer the reader to Bressan and Han (2011) for an in-depth analysis of this model and its variational formulation. The travel cost functions, following Sect. 5.1.2, are  φ(t) = − t,

.

ψ(t) =

0

if t < 0

t2

if t ≥ 0

Here, we have omitted the subscripts i and j since there is only one O-D pair; .φ(·) is the cost associated with departure; and a later departure always means less travel cost. The arrival cost function, .ψ(·), stipulates that no penalties are imposed if the arrival time is less than .0 ∈ [t0 , tf ], which is defined to be the target arrival time for simplicity. And the penalty for late arrival grows quadratically. The notion of dynamic user equilibrium here is given as follows, which differs slightly from Definition 3.1. In essence, this definition of a Nash equilibrium allows path departure rates based on the Dirac-.δ; in other words, the cumulative departure curve can contain jump discontinuities. In this case, the departure rates are no longer square-integrable. We let .H (·) be the cumulative departure curve (departure profile). Obviously the demand satisfaction constraint is expressed as H (t0 ) = 0,

.

H (tf ) = Q

(5.38)

We also let .U (·) be the arrival curve. In order to accommodate this more general class of departure profiles, we first introduce .β ∈ [0, Q] as the Lagrangian label associated with a given driver, where Q is the travel demand. We then define the times .t q (β) and .t a (β) by setting .

t q (β) = sup{t ∈ [t0 , tf ] : H (t) ≤ β} t a (β) = sup{t ∈ [t0 , tF ] : U (t) ≤ β}

5.3 An Example of Non-existence of DUE

175

Here, .t q (β) accounts for the time when the driver labeled as .β joins the queue, and a .t (β) is the arrival time. The following Nash equilibrium among drivers is defined in terms of the cumulative departure curve .H (·), rather than the departure rate. Definition 5.1 We say that a bounded, nondecreasing function .H (·) satisfying (5.38) yields a Nash solution with departure and arrival cost functions .φ and .ψ if there exists a constant c (common cost) such that: 1. For almost every .β ∈ [0, Q] one has     φ t d (β) + ψ t a (β) = c

.

2. For all .t ∈ [t0 , tf ], there holds φ(t) + ψ(A(t)) ≥ c

.

The first condition states that all drivers bear the same cost c. The second condition says that, regardless of the starting time, no one can achieve a cost less than c. Intuitively, this definition of Nash equilibrium coincides with a DUE with departure time choice only, except that it allows a more general definition of the departure profile. Given the fundamental diagram .f (·) and the cost functions .φ(·) and .ψ(·), the Nash equilibrium solution has the following properties (see Fig. 5.1).

x

tS

t

S

τ0

τ2

q

τ (t)

τ3

0

τ

4

τ1

t

τ4

τ1

t

H(t) Q(t) δ0 τ0 Fig. 5.1 The Nash equilibrium solution

176

5 Existence of Dynamic User Equilibria

. • Before time .τ0 = −c, no cars enter the queue. • Exactly at time .τ0 , an amount .δ0 of cars arrives and instantly forms a queue at the entrance of the link. Here, . .δ0 =



0

u(t, ˜ L) dt > 0

t0

where .u˜ = u(t, ˜ x) is the solution of the Riemann problem:  ∂x u˜ + ∂t f (u) ˜ = 0,

.

u(t, ˜ 0) =

0

if t < τ0

C

if t ≥ τ0

and L is the length of the link. • The last of the cars that entered the queue at .t = τ0 departs at time .τ2 = τ0 + (δ0 /C) and arrives at its destination exactly at time .t = 0. • The queue empties at some time .τ3 . When this happens, a shock wave is formed, moving along some curve .S. • After time .τ3 , cars keep coming to the entrance of the link, and depart instantly, until time .τ1 . No drivers begins his/her journey after time .τ1 . Here, %  $   L L L 2 1 1 . +c− ≤ c = − + .τ1 = sup t : − t + t + v0 2 v0 4 v0 The defining characteristic of this Nash departure profile is its “instant” queue forming at .τ0 . It is quite easy to understand why such a queue is necessary: the arrival penalty for .t < 0 is zero, thus those who arrive before time .t = 0 have to depart all at the same time; otherwise, they will experience different departure cost .φ(t) = −t while receiving the same (zero) arrival cost, leading to a nondisequilibrium. It is quite clear that the departure rate corresponding to .H (t) is not an integrable function since .H (t) has a jump discontinuity at time .τ0 (see Fig. 5.1). In addition, Bressan and Han (2011) further show that there is no other Nash equilibrium solution for this problem. Therefore, we conclude that no solution in the sense of Definition 3.1 exists. A closer look at the cause for this non-existence of solutions reveals that condition (5.28) from assumption A1 is violated by the choice of .ψ; that is, for .t0 < t1 < t2 < 0, ψ(t2 ) − ψ(t1 ) = 0 − 0 = 0 < Δ(t2 − t1 )

.

for any .Δ > 0. In other words, the arrival cost function is not strictly increasing. This coincides with our analysis presented earlier: the equal arrival penalty for all arrivals before time 0 causes the instant queue to form, which represents a Dirac-.δ in the departure rate.

5.4 Existence of Dynamic User Equilibrium with Bounded Rationality

177

This section shows that, even in the simplest network setting, the DUE problem may not have a solution. Indeed, this is a result of a singularity introduced by certain forms of the drivers’ cost functions. In particular, we see that the condition A1 is necessary for the existence of a DUE solution. However, as we will see in the next section, such a condition could be further relaxed for the existence of boundedly rational dynamic user equilibria.

5.4 Existence of Dynamic User Equilibrium with Bounded Rationality The existence of dynamic user equilibrium with simultaneous route and departure time choices will be shown in this section, when the notion of bounded user rationality is incorporated. In particular, we will show the existence of variable tolerance boundedly rational dynamic user equilibrium (VT-BR-DUE), where the user tolerance is endogenous and may depend on the O-D pair, the path, and the established (realized) path departure rates. Notice that VT-BR-DUE is the most general notion of DUE incorporating bounded rationality, and its existence immediately implies the existence of BR-DUE (with exogenous tolerances) and route-choice BR-DUE. The existence of a dynamic user equilibrium trivially implies the existence of a boundedly rational dynamic user equilibrium. Therefore, a meaningful discussion of the existence of VT-BR-DUEs, as is our aim in this section, should rely on assumptions weaker than those made for normal DUEs. In particular, we will show the existence of VT-BR-DUE under conditions weaker than those assumed for SRDT DUE (i.e., A1–A3), thereby establishing that the existence of VT-BR-DUE is indeed more general than DUE. In particular, the following assumption will be invoked regarding the cost functions .φij and .ψij : For each .(i, j ) ∈ W, .φij (·) and .ψij (·) are continuous on .[t0 , tf ]; .ψij (·) is monotonically increasing; and .φij (·) is Lipschitz continuous with constant .Lij .

A0.

Clearly, A0 is more general than A1 and A2. The variational inequality (VI) formulation of VT-BR-DUE will play a vital role in our existence theory. We recall that a vector of path departure rates .h∗ ∈ Λ is a VT-BR-DUE if and only if it solves the following VI:  .

Φ ε (h∗ ), h − h∗



≥ 0

∀h ∈ Λ

(5.39)

where the principal operator is defined as:  |P | Φ ε : Λ → L2+ [t0 , tf ] ,

.

  h → Φpε (·, h) : p ∈ P

(5.40)

178

5 Existence of Dynamic User Equilibria

and   p Φpε (t, h) = max Ψp (t, h), vij (h) + εij (h)    p q ∀(i, j ) ∈ W, − εij (h) − min εij (h)

.

q∈Pij

∀p ∈ Pij

(5.41)

Before applying Browder’s existence theorem, we need to better understand the structure of this VI by establishing some analytical properties of the operator .Φ ε . This will be the goal of the next section.

5.4.1 Analytical Properties of the New Operator In this section, we will show that the delay operator .Φ ε , given by (5.40) and (5.41), exists, is well defined, and is continuous. Again, these results will automatically hold for .φ ε . Proposition 5.5 (Existence and well-definedness of .Φ ε as an operator) Let  2 |P | .Ψ : Λ → L+ [t0 , tf ] be the effective path delay operator, and .ε(·) =  p  εij (·), p ∈ Pij , (i, j ) ∈ W be the variable tolerance. Assume that the tolerances are uniformly bounded; that is,   p 0 < sup εij (h) : p ∈ Pij , (i, j ) ∈ W, h ∈ Λ < + ∞

.

(5.42)

Then the operator .Φ ε , given by (5.40) and (5.41), exists and is a well-defined |P |  . mapping from .Λ to . L2+ [t0 , tf ] Proof Given any .h ∈ Λ, by virtue  of the effective path delay operator there exists a unique vector-valued function . Ψp (·, h), p ∈ P of t. Moreover, such a vector |P | valued function belongs to the set . L2+ [t0 , tf ] . Then, according to (5.41) there exists a unique vector-valued function . Φpε (·, h), p ∈ P of t. In addition, due to p (5.42) there exists an upper bound .M < +∞ for all the functionals .εij (·), p ∈ Pij , (i, j ) ∈ W. Then we have, for every .p ∈ P, that 

tf

.

t0

  2 ε Φp (t, h) dt ≤

tf



Ψp (t, h) + M

2

dt

t0

 =

tf

 2 Ψp (t, h) dt

t0

+ 2M



tf t0

Ψp (t, h) dt + (tf − t0 )M 2 < + ∞

5.4 Existence of Dynamic User Equilibrium with Bounded Rationality

179

Here we have used the fact that a square-integrable function on a compact set is also  |P | integrable. Therefore, .Φ ε (h) ∈ L2+ [t0 , tf ] . We conclude that the operator .Φ ε exists and is well defined.  Notice that Proposition 5.5 is based on the well-accepted premise that an effective path delay operator exists. Rigorous existence results for the effective path delay operators are presented by Friesz et al. (1993) for the DNL based on the link delay model; by Han et al. (2013c) for the DNL based on the Vickrey model; by Bressan and Han (2012b) and Friesz et al. (2013) for the DNL based on the LWR-Lax model; and by Garavello and Piccoli (2006) for the DNL based on the classical LWR model. For the existence proof, it remains to show the continuity of .Φ ε , which relies on the continuity of the effective path delay operator. The next theorem establishes the continuity of the new operator .Φ ε when viewed as a mapping from .Λ to  2 |P | . L+ [t0 , tf ] . Our argument relies on the already established results regarding the continuity of the effective delay operator .Ψ , more precisely, on assumption A3 stated in Sect. 5.1.3. Theorem 5.4 (Continuity of .Φ ε as an operator) Assume that the effective path  |P |  p and the tolerance .ε = εij (·) : p ∈ delay operator .Ψ : Λ → L2+ [t0 , tf ]  |P | Pij , (i, j ) ∈ W : Λ → R+ satisfies A3. Then the operator  |P | Φ ε : Λ → L2+ [t0 , tf ] ,

.

  h → Φpε (·, h) : p ∈ P

(5.43)

where .∀p ∈ Pij , ∀(i, j ) ∈ W, ε .Φp (t,

     p p q h) = max Ψp (t, h), vij (h) + εij (h) − εij (h) − min εij (h) , q∈Pij

(5.44) also satisfies A3. Before we begin the continuity proof, we make a very simple yet crucial observation regarding the effective path delay .Ψp (·, h), viewed as a function of departure time t. Lemma 5.1 Assume that A0 holds. Then under the first-in-first-out (FIFO) rule, we have Ψp (t2 , h) − Ψp (t1 , h) ≥ − Lij (t2 − t1 )

.

∀p ∈ Pij , ∀h ∈ Λ

(5.45)

for any .t0 ≤ t1 ≤ t2 ≤ tf . Here, .Lij is the Lipschitz constant associated with the function .φij as articulated in assumption A0.

180

5 Existence of Dynamic User Equilibria

Proof According to FIFO, a later departure time implies a later arrival time along the same path; we have, for any .t1 ≤ t2 , that t1 + Dp (t1 , h) ≤ t2 + Dp (t2 , h)

.

∀p ∈ Pij , ∀h ∈ Λ

According to the alternative representation of the effective path delay and assumption A0, we deduce that   Ψp (t2 , h) − Ψp (t1 , h) = φij (t2 ) + ψij t2 + Dp (t2 , h)   − φij (t1 ) − ψij t1 + Dp (t1 , h)   ≥ −Lij (t2 − t1 ) + ψij t2 + Dp (t2 , h)   − ψij t1 + Dp (t1 , h)

.

≥ −Lij (t2 − t1 )  We now begin the proof of Theorem 5.4. Proof The proof is divided into several parts. Part 1.

Consider an arbitrary O-D pair .(i, j ) ∈ W. We show in this part that if any sequence .{h(n) }n≥1 ⊂ Λ, uniformly bounded by a constant, converges weakly to .h∗ ∈ Λ, then the essential infima satisfy .|vij (h(n) )−vij (h∗ )| → 0, where these essential infima are defined in (3.6)–(3.7). According to property A3 of .Ψ , we have that Ψp (t, h(n) ) → Ψp (t, h∗ )

.

n → +∞

uniformly, and thus .Ψ (h(n) ) − Ψ (h∗ )L2 → 0. This means that, given any .σ > 0, there exists .N > 0 such that, for each .n > N, we have σ 3/2 (n) ) − Ψ (h∗ ) , where .Lij is the Lipschitz constant from .Ψ (h L2 < √ 8Lij

A0. For every .n > N, if .vij (h(n) ) ≤ vij (h∗ ), then we claim vij (h(n) ) ≥ vij (h∗ ) − σ

.

(5.46)

We show (5.46) by contradiction. Assume that .vij (h(n) ) < vij (h∗ ) − σ and that .vij (h(n) ) is attained at some time .tˆ ∈ [t0 , tf ] for some path .p ˆ ∈ Pij . Then, for every .t ∈ [tˆ − 2Lσ ij , tˆ], according to (5.45) Ψpˆ (tˆ, h(n) ) − Ψpˆ (t, h(n) ) ≥ − Lij (tˆ − t) ⇒

.

Ψpˆ (t, h(n) ) ≤ Ψpˆ (tˆ, h(n) ) + Lij (tˆ − t)

5.4 Existence of Dynamic User Equilibrium with Bounded Rationality

181

= vij (h(n) ) + Lij (tˆ − t) < vij (h∗ ) − σ + Lij (tˆ − t) ≤ vij (h∗ ) − σ σ 2Lij σ σ = vij (h∗ ) − ≤ Ψpˆ (t, h∗ ) − 2 2 + Lij ·

Thus, we have that .

 σ 3/2 & > Ψ (h(n) ) − Ψ (h∗ )L2 = 8Lij p∈P



tf

t0

' ' 'Ψp (t, h(n) )

− Ψp (t, h∗ )|2 dt ≥



tˆ tˆ− 2Lσ

1/2

' ' 'Ψpˆ (t, h(n) )

ij

'2 1/2 ' − Ψpˆ (t, h∗ )' dt  ≥

σ2 σ · 2Lij 4

1/2

which leads to a contradiction. Thus (5.46) must hold. On the other hand, if .vij (h(n) ) ≥ vij (h∗ ), we may similarly show that vij (h∗ ) ≥ vij (h(n) ) − σ

.

Part 2.

In any case, we must have .|vij (h(n) ) − vij (h∗ )| < σ . We have thus far shown that .|vij (h(n) ) − vij (h∗ )| → 0 for every .(i, j ) ∈ W. Given any uniformly bounded sequence .{h(n) }n≥1 that converges to .h∗ weakly, according to (5.44), for every .t ∈ [t0 , tf ] and every .p ∈ Pij , we have ' ' ε 'Φ (t, h(n) ) − Φ ε (t, h∗ )' p p

.

' ' ' ≤ max 'Ψp (t, h(n) ) − Ψp (t, h∗ )', 'vij (h(n) ) − vij (h∗ ) ' p p + εij (h(n) ) − εij (h∗ )'

!

' p  q  q  ' p + 'εij (h(n) ) − εij (h∗ ) − min εij (h(n) ) + min εij (h∗ ) ' q∈Pij

' ' (n) ≤'Ψp (t, h(n) ) − Ψp (t, h∗ )' + Aij →0 + 0 = 0

uniformly as

n→∞

q∈Pij

182

5 Existence of Dynamic User Equilibria

' ' where .'Ψp (t, h(n) ) − Ψp (t, h∗ )' → 0 uniformly is due to A3, and . '' p (n) p ∗ '' (n) ∗ A(n) ij = vij (h ) − vij (h ) + εij (h ) − εij (h ) ' p  q  p + 'εij (h(n) ) − εij (h∗ ) − min εij (h(n) )

.



+ min

q∈Pij

' q εij (h∗ ) '

q∈Pij

→ 0 p

due to the result established in Part 1 and the fact that each .εij (·) satisfies A3, .p ∈ Pij , (i, j ) ∈ W. This shows that .Φpε (t, h(n) ) converges uniformly to .Φp (t, h(n) ) for all .t ∈ [t0 , tf ] and .p ∈ P, thereby establishing the A3 property for .Φ ε .  Remark 5.3 Theorem 5.4 highlights the fact that the continuity of .Φ ε is more general than the continuity of the effective path delay operator .Ψ , simply because the latter implies the former. The reverse is false, and one can easily envisage situations where the effective delay operator is discontinuous but the corresponding ε .Φ are continuous. Consider, for example, a uniformly bounded, weakly convergent sequence .h(n) → h∗ but . lim Ψ (h(n) ) = Ψ (h∗ ). If the tolerance .ε is large enough n→+∞

to cover such a jump discontinuity, the functions .Φ ε (h(n) ) can still converge to ∗ .Φ(h ) uniformly. A more concrete example is provided below. Example 5.1 In Part 1 of Fig. 5.2, we show the sequence .Ψp (·, h(n) ) and their limit (n) . lim Ψp (·, h ), where each function consists of three components, labeled by I, n→∞ II, and III in the figure. Moreover, these functions share the same components I and I I I and differ only in I I . In Part 2, the function .Ψp (·, h∗ ) is shown with thick line segments, and it is different from . lim Ψp (·, h(n) ), indicating a discontinuity n→∞

of .Ψ at the point .h∗ . In Part 3, if all the differences in these functions are within the indifference band, .Φpε (·, h(n) ), lim Φpε (·, h(n) ), and .Φpε (·, h∗ ) would coincide. n→∞

As a result, .Φ ε is continuous at .h∗ .

p

(t, h (n) )

p

p

(t, h (n) )

III

III

I

(t, h (n) )

I

II

vij+

II

vij n=1

p

n=2 n=3 n=

Part 1

t

Part 2

(t, h* )

t

Fig. 5.2 Illustration that .Φ ε may be continuous even if .Ψ is not

p

Part 3

(t, h* )

t

5.4 Existence of Dynamic User Equilibrium with Bounded Rationality

183

This example shows that the continuity of .Φ ε is indeed a weaker regularity condition than the continuity of .Ψ .

5.4.2 The Existence Proof We present our main existence result for the VT-BR-DUE problem in the following theorem. Theorem 5.5 (Existence of VT-BR-DUE) Assume that 1. Assumption A0 holds.  |P | 2. The operator .Φ ε : .Λ → L2+ [t0 , tf ] is continuous in the sense that it satisfies A3. p 3. The functionals .εij (·) : Λ → R+ satisfy A3 and are bounded away from p zero. That is, there exists .εmin > 0 such that .εij (h) ≥ εmin , ∀h ∈ Λ, ∀p ∈ Pij , ∀(i, j ) ∈ W. Then, the variable tolerance boundedly rational dynamic user equilibrium with simultaneous route and departure time choices, as defined in Definition 3.4, exists. Proof We consider, for each natural number .n ≥ 1, a uniform partition of the compact interval .[t0 , tf ] into n sub-intervals .I1 , . . . , In with the size of each being .(tf − t0 )/n. We then consider the following subsets: .  Λn = h ∈ Λ : hp (·) is constant on Ii ,

.

∀i = 1, . . . , n,

∀p ∈ P



⊂ Λ ∀n ≥ 1 Notice that each .Λn is the intersection of .Λ and the space of piecewise constant functions. Since .Λ is expressed via linear constraints, each set .Λn is clearly convex. In addition, due to the finite-dimensional nature of .Λn , it is also compact. A detailed proof of compactness for .Λn based on sequential compactness has been presented in the proof of Theorem 5.2 and will be omitted here. Following the same proof of Theorem 5.2, we can show that .Φ ε , for which A3 is  |P | true, is strongly continuous from .Λn into . L2+ [t0 , tf ] . Theorem 5.1 then states that for each .n ≥ 1, there exists .hn,∗ ∈ Λn such that for all .hn ∈ Λn ,  .

 Φ ε (hn,∗ ), hn − hn,∗ ≥ 0 or

 p∈P

tf

t0

  Φpε (t, hn,∗ ) hnp (t) − hn,∗ (t) dt ≥ 0 p (5.47)

184

5 Existence of Dynamic User Equilibria

Since both .hn and .hn,∗ are piecewise constant, it follows from (5.47) that for any .(i, j ) ∈ W,  hn,∗ p (t) > 0, t ∈ Ik ⇒

.

Ik

 Φpε (t, hn,∗ ) dt = min

min

q∈Pij l=1,...,n Il

Φqε (t, hn,∗ ) dt (5.48)

for all .p ∈ Pij and .k = 1, . . . , n. In other words, within the same origin-destination pair, the integrals of the function .Φpε (·, hn,∗ ) for all utilized paths and departure time intervals are equal and minimal. In view of the first and third assumptions, we choose .n ≥ 1 such that the size min . t −t of the subintervals, .δn = 0 n f , satisfies .δn < 2 max(i,ε j )∈W Lij . Fixing any origindestination pair .(i, j ) ∈ W, we denote by .vij (hn,∗ ) the essential infimum of the path effective delays, which is attained at some point .tˆ ∈ Il corresponding to some path .q ∈ Pij . Clearly, by taking the time horizon large enough one can always assume that .l > 1. That is, .Il is not the first time interval because the corresponding early arrival penalty would be very large. Then we consider the interval .Il−1 and deduce the following based on (5.45), which is a consequence of A0, Ψq (tˆ, hn,∗ ) − Ψq (t, hn,∗ ) ≥ − Lij (tˆ − t)

.

⇒Ψq (t, hn,∗ ) ≤ vij (hn,∗ ) + Lij (tˆ − t) ≤ vij (hn,∗ ) + 2Lij δn ≤ vij (hn,∗ ) + εmin

(5.49)

for all .t ∈ Il−1 . We immediately have that ε .Φq (t,

h

n,∗

) = vij (h

n,∗

q ) + εij (hn,∗ ) − q

= vij (hn,∗ ) + min {εij } q ∈P

  q q n,∗ εij (h ) − min {εij } q ∈Pij

∀t ∈ Il−1

ij

For any .p ∈ Pij and any interval .Ik such that .hn,∗ p (t) > 0, .t ∈ Ik , (5.48) implies that     q . Φpε (t, hn,∗ ) dt ≤ Φpε (t, hn,∗ ) dt = δn · vij (hn,∗ ) + min {εij } Ik

Il−1

q ∈Pij

(5.50)

5.5 Characterization of Solutions for Dynamic User Equilibrium with. . .

185

On the other hand, by definition   p p q Φpε (t, hn,∗ ) ≥ vij (hn,∗ ) + εij (hn,∗ ) − εij (hn,∗ ) − min {εij }

.

q ∈Pij

q

= vij (hn,∗ ) + min {εij } q ∈Pij

(5.51)

Equations (5.50) and (5.51) together imply that .Φpε (t, hn,∗ ) = vij (hn,∗ ) + q

minq ∈Pij {εij } for almost every .t ∈ Ik . Finally, by definition (5.44), such an p equality holds if and only if .Ψp (t, hn,∗ ) ≤ vij (hn,∗ ) + εij (hn,∗ ) for almost every n,∗ is a VT-BR.t ∈ Ik . Since .(i, j ), p and .Ik are arbitrary, we have established that .h  DUE. Remark 5.4 Theorem 5.5 is significant in that it relaxes all three conditions A1– A3 required for the existence of normal DUEs (see Theorem 5.2). First of all, compared to A1, A0 drops the assumptions that .φij (·) is monotonically decreasing and thereby inequality (5.28). Secondly, assumption A2 is completely omitted in this theorem. Last but not least, Theorem 5.5 relies on the continuity of .Φ ε instead of the continuity of .Ψ . According to Remark 5.3, the former is more general than the latter. As a side note, the third assumption from Theorem 5.5 is a reasonable assumption p necessary for the existence of VT-BR-DUE. Otherwise, if we allow .εij (h(n) ) to tend to zero for a sequence of points .h(n) , then we are back to the normal DUE case and cannot expect a more general existence result to hold.

5.5 Characterization of Solutions for Dynamic User Equilibrium with Bounded Rationality This section provides a mathematical characterization of the solution set of the VTBR-DUE problems. Theorem 5.5 shows that such a set is non-empty under mild conditions, which will continue to hold throughout this section. We will restrict our analysis to a finite-dimensional space (i.e., discrete-time problems) for the following reasons: (1) it is easier to describe and depict various geometric and topological properties in a finite-dimensional space, which is effectively an Euclidean space, and (2) working with finite-dimensional cases sheds light on the numerical solutions of VT-BR-DUEs as almost all calculations are done in a discrete-time setting. The discrete-time VT-BR-DUE problem will be rigorously defined below.

186

5 Existence of Dynamic User Equilibria

5.5.1 Discrete-Time VT-BR-DUE Problem We consider, for each .n ≥ 1, a uniform partition of .[t0 , tf ] into n subintervals I1 , . . . , In . Define

.

.  Λn = h ∈ Λ : hp (·) is constant on Ik ,

∀1 ≤ k ≤ n,

.

∀p ∈ P



∀n ≥ 1 (5.52)

Each .Λn is the intersection of .Λ and the space of piecewise-constant functions. Notice that each .Λn still consists of continuous-time functions. We next introduce the discrete-time counterpart of the departure rate vectors. Given .n ≥ 1 and .h ∈ Λn , define a vector .h¯ p ∈ Rn+ such that .h¯ p (k) = hp (t), t ∈ Ik for all .1 ≤ k ≤ n. In other  words, .h¯ p is the discrete-time departure rate. Naturally, we let .h¯ = h¯ p , p ∈ P ∈ n×|P |

R+ be the vector of all the discrete-time departure rates. We define the feasible set of the discrete-time path departure rates as ⎧ ⎨

. ¯n = .Λ

⎩

h¯ ∈

n×|P | R+

: δt

n 

h¯ p (k) = Qij

∀(i, j ) ∈ W

p∈P k=1

⎫ ⎬ ⎭

(5.53)

where .δt is the time step size. We will next define the effective path delays in discrete time. Given .h ∈ Λn , the corresponding effective path delays .Ψp (·, h), p ∈ P is, in general, not piecewise constant. We let  1 Ψp (t, h) dt ∀1 ≤ k ≤ n, ∀p ∈ P . |Ik | Ik be the average value of the effective delay on path p corresponding to departure interval .Ik , where .|Ik | is the length of .Ik . Let us define the discrete-time effective path delay operator according to the following chain of mappings   h¯ ∈ Λ¯ n → h ∈ Λn → Ψp (·, h), p ∈ P    1 Ψp (t, h) dt, 1 ≤ k ≤ n, p ∈ P , → |Ik | Ik

.

which defines a mapping n×|P | Ψ¯ : Λ¯ n → R+ ,

.

  ¯ 1 ≤ k ≤ n, p ∈ P h¯ → Ψ¯ p (k, h),

where . ¯ = Ψ¯ p (k, h)

.

1 |Ik |

 Ψp (t, h) dt Ik

(5.54)

5.5 Characterization of Solutions for Dynamic User Equilibrium with. . .

187

Ψ¯ is the discrete-time counterpart of the effective path delay operator. With these preliminaries, we are now ready to introduce discrete-time versions of the DUE and VT-BR-DUE problems.

.

Definition 5.2 (Discrete-time DUE problem) For each .n ≥ 1, a vector of path departure rates .h¯ ∗ ∈ Λ¯ n is a solution of the finite-dimensional DUE problem if for any .(i, j ) ∈ W, we have h¯ ∗p (k) > 0, p ∈ Pij ⇒ Ψ¯ p (k, h¯ ∗ ) = vij (h¯ ∗ )

.

where .vij (h¯ ∗ ) =

min

1≤k≤n, p∈Pij

∀1 ≤ k ≤ n

(5.55)

Ψ¯ p (k, h¯ ∗ ).

It is not difficult to see that the problem defined above is equivalent to the following finite-dimensional variational inequality: 

n 

.

find h¯ ∗ ∈ Λ¯ n such that Ψ¯ p (k, h¯ ∗ )(h¯ p (k) − h¯ ∗p (k)) ≥ 0

p∈P k=1

∀ h¯ ∈ Λ¯ n

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

  V I Ψ¯ , Λ¯ n , [t0 , tf ]

(5.56)

The discrete-time VT-BR-DUE problem is similarly defined. To do this, we let p ε¯ ij (·), p ∈ Pij , (i, j ) ∈ W be a set of mappings from .Λ¯ n into .R++ , such that p ¯ p .ε ¯ ij (h) = εij (h), where .h¯ ∈ Λ¯ n is the discrete-time counterpart of .h ∈ Λn .  p Definition 5.3 (Discrete-time VT-BR-DUE problem) Given .ε¯ = ε¯ ij (·) : p ∈  Pij , (i, j ) ∈ W , a vector .h¯ ∗ ∈ Λ¯ n is a solution of the discrete-time VT-BR-DUE problem if, for any .(i, j ) ∈ W, we have .

  p h¯ ∗p (k) > 0, p ∈ Pij ⇒ Ψ¯ p (k, h¯ ∗ ) ∈ vij (h¯ ∗ ), vij (h¯ ∗ ) + ε¯ ij (h¯ ∗ )

.

where .vij (h¯ ∗ ) =

min

1≤k≤n, p∈Pij

(5.57)

Ψ¯ p (k, h¯ ∗ ).

The following finite-dimensional VI characterizes the solution of the discrete-time VT-BR-DUE problem: n  .

p∈P k=1

find h¯ ∗ ∈ Λ¯ n such that Φ¯ pε¯ (k, h¯ ∗ )(h¯ p (k) − h¯ ∗p (k)) ≥ 0 ∀ h¯ ∈ Λ¯ n

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

  V I Φ¯ ε¯ , Λ¯ n , [t0 , tf ]

(5.58)

188

5 Existence of Dynamic User Equilibria

where for all .1 ≤ k ≤ n, p ∈ Pij , we have      p p q Φ¯ pε¯ (k, h¯ ∗ ) = max Ψ¯ p (k, h¯ ∗ ), vij (h¯ ∗ ) + ε¯ ij (h¯ ∗ ) − ε¯ ij (h¯ ∗ ) − min ε¯ ij (h¯ ∗ ) ,

.

q∈Pij

(5.59)  n×|P | ¯ ¯ 1 ≤ k ≤ n, p ∈ which defines the operator .Φ¯ ε¯ : Λ¯ n → R+ , .h → Φ¯ pε¯ (k, h),  P . The following lemma regarding the continuity of the discrete-time operators is useful as it substantiates the assumptions made in subsequent analysis.  |P | p Lemma 5.2 (Continuity of .Ψ¯ and .Φ¯ ε¯ ) If .Ψ : Λ → L2+ [t0 , tf ] and .εij (·) : n×|P | Λ → R+ both satisfy A3 (see Sect. 5.2.2), then .Ψ¯ : Λ¯ n → R+ and .Φ¯ ε¯ : Λ¯ n → n×|P |

R+

are both continuous.

Proof For each .n ≥ 1, fix any point .h¯ ∈ Λ¯ n , and consider an arbitrary sequence ¯ (m) ∈ Λ¯ n , .m ≥ 1, that converges to .h¯ in the Euclidean norm. We let .h ∈ Λn .h and .h(m) ∈ Λn be the continuous-time counterparts of .h¯ and .h¯ (m) , respectively. It  |P | and thus weakly. is easy to verify that .h(m) → h in the .L2 -norm in . L2 [t0 , tf ] (m) In addition, the sequence .{h }m≥1 is uniformly bounded. Thus invoking A3 for .Ψ yields .

# #2 # # #Ψ (h) − Ψ (h(m) )# 2 → 0, L

as m → +∞

Next, in order to show that .Ψ¯ is continuous, it suffices to verify that .Ψ¯ p (k, h¯ (m) ) → ¯ for every .p ∈ P and .1 ≤ k ≤ n, as .m → ∞. Indeed, recalling from (5.54) Ψ¯ p (k, h) that ' ' ' ' ' 1 '' ' '¯ (m) (m) ¯ ¯ ¯ p (k, h )' = . 'Ψp (k, h) − Ψ Ψp (t, h) − Ψp (t, h ) dt '' ' |Ik | Ik  ' ' 1 ' ' ≤ 'Ψp (t, h) − Ψp (t, h(m) )' dt |Ik | Ik  ' 1 & 'Ψp (t, h) ≤ · |Ik | · |Ik | Ik '2  12 (m) ' − Ψp (t, h )' dt → 0 as .m → ∞, for every k and every p. Notice that the last inequality is a consequence of Jensen’s inequality. This establishes the continuity of .Ψ¯ .

5.5 Characterization of Solutions for Dynamic User Equilibrium with. . .

189

To see the continuity of .Φ¯ ε¯ , we notice that the continuity (A3) of functionals ∈ Pij , (i, j ) ∈ W immediately leads to the continuity of the functions definition. The rest of the proof simply follows from (5.59). 

p .ε (·), .p ij p .ε ¯ ij (·) by

5.5.2 Characterization of the Solution Set of VT-BR-DUE We begin with the first theorem that characterizes the solution set as being closed and bounded in the finite-dimensional Euclidean space .Rn×|P | . Proposition 5.6 (Compactness of the solution set) Let .Φ¯ ε¯ be continuous as shown in Lemma 5.2, then all the solutions of the discrete-time VT-BR-DUE problem form a closed and bounded, and thus compact, set in .Λ¯ n ⊂ Rn×|P | for every .n ≥ 1. Proof We let .h¯ (m) ∈ Λ¯ n , .m ≥ 1 be an arbitrary sequence of solutions that converge n×|P | to some .h¯ ∗ ∈ R+ . Moreover, for every .(i, j ) ∈ W, δt

n  

.

h¯ ∗p (k) = δt

p∈Pij k=1

n   p∈Pij k=1

=

lim δt

m→+∞

lim h¯ (m) (k) m→+∞ p

n  

h¯ (m) p (k) = Qij

p∈Pij k=1

where .δt is the time step size. This means that .h¯ ∗ ∈ Λ¯ n . We will next show that .h¯ ∗ is a solution of the VT-BR-DUE problem by proving that it satisfies the VI (5.58). Indeed, for any .h¯ ∈ Λ¯ n , we have that n  .

  Φ¯ pε¯ (k, h¯ (m) ) h¯ p (k) − h¯ (m) p (k) ≥ 0

∀m ≥ 1

p∈P k=1

According to the continuity result provided by Lemma 5.2, we deduce that 0 ≤

.

=

lim

m→+∞

n  p∈P k=1

n  p∈P k=1

=

  Φ¯ pε¯ (k, h¯ (m) ) h¯ p (k) − h¯ (m) p (k)

n  p∈P k=1

  lim Φ¯ pε¯ (k, h¯ (m) ) h¯ p (k) − h¯ (m) p (k)

m→+∞

  Φ¯ pε¯ (k, h¯ ∗ ) h¯ p (k) − h¯ ∗p (k)

190

5 Existence of Dynamic User Equilibria

for all .h¯ ∈ Λ¯ n . Thus .h¯ ∗ is a solution. We have thus shown that the set of discretetime VT-BR-DUE solutions is closed. To see that the set is also bounded, we have the following estimate for a given .n ≥ 1: h¯ p (k) ≤

.

max(i, j )∈W Qij (tf − t0 )/n

∀1 ≤ k ≤ n, ∀p ∈ P

for all .h¯ ∈ Λ¯ n . In other words, the vectors in .Λ¯ n are element-wise uniformly bounded; thus their norms are also uniformly bounded.  Having established that for each .n ≥ 1, the discrete-time solution set, denoted by Ω n , is compact, we would like to further characterize its interior points. An interior point of a set is such that there exists a ball centered at this point that is completely contained in the set. In other words, if a solution .h¯ ∗ ∈ Ω n is an interior point, then one can perturb such a point in any direction by a small amount and find another solution. Additionally, we say that a point x is an interior point of a set S relative to another set .X ⊃ S if there exists a ball .Bxδ centered at x with radius .δ > 0 such that δ .Bx ∩ X ⊂ S. .

Example 5.2 (Non-existence of interior points of .Ω n relative to the whole space) We first observe that .Ω n does not have any interior point relative to the whole space n×|P | . This is because any ball .B δ ⊂ Rn×|P | centered at a solution .h∗ ∈ Ω n .R h¯ ∗ with radius .δ > 0 obviously contains points that violate the demand satisfaction constraints (5.53) and hence do not belong to the set .Ω n . The next question we would like to ask is whether .Ω n has any interior points relative to .Λ¯ n . Again, the answer is no, meaning that a point .h¯ ∈ Λ¯ n arbitrarily close to a solution .h¯ ∗ may fail to be a solution. This will be illustrated in the following example. Example 5.3 (Non-existence of interior points of .Ω n relative to .Λ¯ n ) We consider a network consisting of just one path p. Let the time horizon .[t0 , tf ] be large enough such that for any solution .h¯ ∗p ∈ Ω n , the effective path delay corresponding to the first time interval is far greater than the experienced effective path delays for utilized departure intervals. That is, h¯ ∗p (k) > 0 ⇒ Ψ¯ p (k, h¯ ∗p )  Ψ¯ p (1, h¯ ∗p )

.

This corresponds to the situation where a driver departs very early, say at 2am in the morning when the target arrival time at work is 8am; as a result, he/she experiences a great travel cost even though there is little or no congestion. See Fig. 5.3 for an illustration of this situation. To see that a point in .Λ¯ n close enough to .h¯ ∗p may not be a VT-BR-DUE solution, we simply move some traffic from the departure window of .h¯ ∗p to the first time ¯n interval and call the resulting departure pattern .h¯ ∗∗ p , which clearly belongs to .Λ but

5.5 Characterization of Solutions for Dynamic User Equilibrium with. . .

-

- (k, h- * ) p

h*p (k)

n

1

k

-

h** p (k)

191

- (k, h- ** ) p

n

1

k

Fig. 5.3 An illustration that .Ω n does not have an interior point when viewed as a subset of .Λ¯ n (i.e., a point in .Λ¯ n arbitrarily close to a solution may fail to be a solution). One can move some traffic from the departure window of a solution .h¯ ∗ to the first and the resulting departure # time interval, # # ¯∗ ¯ ∗∗ # ¯n profile .h¯ ∗∗ p ∈ Λ is not a VT-BR-DUE, despite that .#hp − hp # can be arbitrarily small. Here 2 . · 2 denotes the standard 2-norm in Euclidean spaces

is not a VT-BR-DUE solution, despite the fact that it differs from .h¯ ∗p by as little as one wants. This shows that .Ω n has no interior points when viewed as a subset of ¯ n. .Λ Example 5.3 suggests that, starting from a given solution, even when the search is restricted to points within the feasible set .Λ¯ n , one cannot always find a solution of the VT-BR-DUE problem. Clearly, additional constraints for the search direction are needed to guarantee one or more solutions are found. The rest of this section is dedicated to the articulation of such conditions and the procedure for finding infinitely many solutions. The following concept turns out to be crucial. Definition 5.4 A discrete-time VT-BR-DUE solution .h¯ ∗ is said to have the (P) property if, for all .(i, j ) ∈ W, we have p h¯ ∗p (k) > 0, p ∈ Pij ⇒ Ψ¯ p (k, h¯ ∗ ) < ε¯ ij (h¯ ∗ )

.

. where .vij (h¯ ∗ ) =

min

1≤k≤n, p∈Pij

∀1 ≤ k ≤ n

Ψ¯ p (k, h¯ ∗ ) is the minimum effective delay between

(i, j ).

.

The (P) property means that in a VT-BR-DUE solution no driver experiences a p travel cost that reaches his/her maximum tolerance (i.e., .vij (h¯ ∗ ) + ε¯ ij (h¯ ∗ )). As a special case of VT-BR-DUE, all normal DUE solutions satisfy the (P) property. As we shall show later in Proposition 5.7, there are in fact infinitely many solutions with the (P) property. We fix an ordering of the finite set .{(Ik , p) : 1 ≤ k ≤ n, p ∈ P}, which is viewed as a bijective mapping .O: .{(Ik , p) : 1 ≤ k ≤ n, p ∈ P} → {1, 2, . . . , n× |P|}. Simply put, .O assigns a label between 1 and .n × |P| to any pair .(Ik , p), where ¯ ∗ ∈ Ω n that satisfies the (P) .Ik is some time interval and p is some path. For any .h

192

5 Existence of Dynamic User Equilibria

property, we define . F(h¯ ∗ ) =



.

p O(Ik , p) : Ψ¯ p (k, h¯ ∗ ) < vij (h¯ ∗ ) + ε¯ ij (h¯ ∗ ), p ∈ Pij



⊂ {1, 2, . . . , n × |P|} In prose, .F(h¯ ∗ ) identifies the pairs .(Ik , p) whose corresponding effective path delays are strictly less than the maximum tolerable cost. Due to the fact that .h¯ ∗ satisfies the (P) property, .F(h¯ ∗ ) is nonempty. We have the following crucial result. Proposition 5.7 Fix .n ≥ 1, let the effective path delay operator .Ψ¯ and the p mappings .ε¯ ij (·), p ∈ Pij , (i, j ) ∈ W be continuous. The following hold. 1. There exists at least one solution that satisfies the (P) property. 2. For every solution .h¯ ∗ that satisfies the (P) property, there exists .δ > 0 such that   Bhδ¯ ∗ ∩ Λ¯ n ∩ span el : l ∈ F(h¯ ∗ ) ⊂ Ω n

.

 n×|P |   where . el l=1 is the natural basis of .Rn×|P | , and .span el : l ∈ F(h¯ ∗ ) is the ¯∗ linear subspace spanned by vectors .el , .l ∈ F(  h ). δ n ∗ ¯ ¯ 3. The set .Bh¯ ∗ ∩ Λ ∩ span el : l ∈ F(h ) is infinite, and all points in this set have the (P) property. Proof (i) According to assumption 3 in Theorem 5.5, there exists .εmin > 0 such p that .εij (h) ≥ εmin for all .h ∈ Λ. Such a property easily transfers to the finitep ¯ dimensional counterpart by definition; that is, .ε¯ ij (h) ≥ εmin holds for all .h¯ ∈ Λ¯ n and .p ∈ Pij , .(i, j ) ∈ W. Fix any number .0 < δ < εmin , we define a new p set of tolerance functions .ε˜ ij (·) : Λ¯ n → R++ such that ¯ = ε¯ (h) ¯ −δ ε˜ ij (h) ij

.

p

p

∀h¯ ∈ Λ¯ n ,

∀p ∈ Pij ,

∀(i, j ) ∈ W

Clearly, these new tolerance functions are continuous. Hence the mapping .Φ¯ ε˜ , p p defined via (5.59) by replacing .ε¯ ij (·) with .ε˜ ij (·), is continuous as well. We then apply Browder’s existence theorem to obtain a .h¯ ∗ ∈ Λ¯ n such that n  .

  Φ¯ pε˜ (k, h¯ ∗ ) h¯ p (k) − h¯ ∗p (k) ≥ 0

∀h¯ ∈ Λ¯ n

p∈P k=1

The VI above clearly leads to the following statement: for all .(i, j ) ∈ W, p p h¯ ∗p (k) > 0, p ∈ Pij ⇒ Ψ¯ p (k, h¯ ∗ ) < vij (h¯ ∗ )+˜εij (h¯ ∗ ) < vij (h¯ ∗ )+¯εij (h¯ ∗ )

.

5.5 Characterization of Solutions for Dynamic User Equilibrium with. . .

193

where .vij (h¯ ∗ ) is the minimum effective delay within .(i, j ). Thus .h¯ ∗ is a solution with the (P) property. (ii) Let .h¯ ∗ be a solution with the (P) property. We define . σ =

min

.

O(Ik , p)∈F (h¯ ∗ )

  p vij (h¯ ∗ ) + ε¯ ij (h¯ ∗ ) − Ψ¯ p (k, h¯ ∗ ), p ∈ Pij > 0 (5.60)

According to the continuity of .Ψ¯ (·) and .ε¯ ij (·), there exists a .δ > 0 such that # # whenever .#h¯ − h¯ ∗ #2 < δ we have p

.

# # #Ψ¯ (h) ¯ − Ψ (h¯ ∗ )# < σ/3, 2

' ' ' p ¯ ' p 'ε¯ ij (h) − ε¯ ij (h¯ ∗ )' < σ/3

∀p ∈ Pij , ∀(i, j ) ∈ W

(5.61)

where . · 2 is the Euclidean norm. Fix any point .h¯ ∈ Bhδ¯ ∗ ∩ Λ¯ n ∩   span #el : l ∈#F(h¯ ∗ ) . We show that .h¯ is a solution. First of all, notice that since .#h¯ − h¯ ∗ #2 < δ, we must have that .

' ' 'Ψ¯ p (k, h) ¯ − Ψ¯ p (k, h¯ ∗ )' # # ¯ − Ψ¯ (h¯ ∗ )# < σ/3 ≤ #Ψ¯ (h) 2

∀1 ≤ k ≤ n, ∀p ∈ P

(5.62)

Consequently, we must also have .

' ' 'vij (h) ¯ − vij (h¯ ∗ )' < σ/3

∀(i, j ) ∈ W

(5.63)

  Given the fact that .h¯ ∈ span el : l ∈ F(h¯ ∗ ) , for any .(i, j ), we have p h¯ p (k) > 0, p ∈ Pij ⇒ Ψ¯ p (k, h¯ ∗ ) < vij (h¯ ∗ ) + ε¯ ij (h¯ ∗ )

.

(5.64)

A consequence of the right-hand side of (5.64), along with (5.60)–(5.63), is that ¯ < σ/3 + Ψ¯ p (k, h¯ ∗ ) ≤ σ/3 + vij (h¯ ∗ ) + ε¯ p (h¯ ∗ ) − σ Ψ¯ p (k, h) ij

.

¯ + σ/3 + ε¯ p (h) ¯ −σ ≤ σ/3 + σ/3 + vij (h) ij ¯ + ε¯ p (h) ¯ = vij (h) ij We have established the following: ¯ < vij (h) ¯ + ε¯ p (h) ¯ ∀(i, j ) ∈ W, h¯ p (k) > 0, p ∈ Pij ⇒ Ψ¯ p (k, h) ij (5.65)

.

194

5 Existence of Dynamic User Equilibria

which shows that .h¯ p (k) is a solution withthe (P) property. (iii) We note that the set .Bhδ¯ ∗ ∩ Λ¯ n ∩ span el : l ∈ F(h¯ ∗ ) is infinite and, by (5.65), every point in this set has the (P) property.  In Proposition 5.7, each statement is logically dependent on the preceding statement. We thus present them in the way they are despite the fact that the third statement makes the first one redundant. We interpret Proposition 5.7 as follows. It first ensures the existence of at least one solution with the (P) property. Then, given a solution .h¯ ∗ with the (P) property, one can find infinitely many VT-BR-DUE  solutions by searching nearby points in the subset .Λ¯ n ∩ span el : l ∈ F(h¯ ∗ ) . As an immediate corollary, we have the   following result by observing that the set .Bhδ¯ ∗ ∩ Λ¯ n ∩ span el : l ∈ F(h¯ ∗ ) is infinite and convex. Corollary 5.1 For every solution .h¯ ∗ that satisfies the (P) property, there exists an infinite and convex solution set in .Rn×|P | that contains .h¯ ∗ . We use a three-dimensional space to illustrate the structure of the solution set, although there is no fundamental difficulty to extending what is visualized to a very high-dimensional space. As shown in Fig. 5.4, only three dimensions are explicitly plotted. The set .Λ¯ n is convex and shown as the triangular set (which is analogous to a simplex in the three-dimensional space). Let .h¯ ∗ be a solution with the (P) property. Assume that .F(h¯ ∗ ) = {1, 2}, then the point .h¯ ∗ lies in the plane spanned by .e1 and .e2 . The convex set containing VT-BR-DUE solutions, including .h¯ ∗ itself, is highlighted as the red line segment. All relevant solutions are within .δ-distance from ¯ ∗. .h

enx|P|

Fig. 5.4 A visualization of the solution set by searching in the neighborhood of .h¯ ∗ along the direction determined as the intersection of .Λ¯ n and .span{e1 , e2 }

-n

h* e2 e1 span{e1 , e2}

5.5 Characterization of Solutions for Dynamic User Equilibrium with. . .

195

5.5.3 Constructing Connected Subset of the Solution Set Proposition 5.7 suggests a way of expanding the solution set based on a given solution .h¯ ∗ , obtained possibly through a particular computational algorithm. In this section we will extend such a technique to obtain connected subsets of the solution set. To fix the idea, we start with a given solution .h¯ 1 with the (P) property. According to the proof of item 1 from Proposition 5.7, such a point can be found by solving a modified VT-BR-DUE problem. We then search locally for more solutions that satisfy the (P) property according to the procedure described in Proposition 5.7 and call the resulting infinite and convex solution set .S(h¯ 1 ). For every .h¯ 2 ∈ S(h¯ 1 ), we repeat the same procedure to find .S(h¯ 2 ). Such a process will continue until no more points can be included in the solution set. This procedure is illustrated in Fig. 5.5. Once the procedure stops, we call the set of solutions obtained in this way the child set of .h¯ ∗ , denoted by .C(h¯ ∗ ). The next proposition shows that any child set is connected. We begin with a precise mathematical notion of connectedness.

-

(h 5 )

-

h5

-

h4

-

h3

-

-

h1

-

(h 4 )

h2

-

(h1 )

Fig. 5.5 Constructing connected subsets of the solution set .Ω n . Each connected subset is generated from a single solution with the (P) property

196

5 Existence of Dynamic User Equilibria

Definition 5.5 (Connected set) A connected set is a set that cannot be divided into two nonempty subsets which are open in the relative topology Given a topological space X and a subset .S ⊂ X, a subset of S is open in the relative topology if and only if it is an intersection of S with an open set in X. Proposition 5.8 Let .h¯ ∗ be a solution with the (P) property, then .C(h¯ ∗ ) is connected. Proof By contradiction, if .C(h¯ ∗ ) is not connected, then there exist two subsets, A and B, of .C(h¯ ∗ ) such that .A ∩ B = ∅ and .A ∪ B = C(h¯ ∗ ). Moreover, both A and B are open in the relative topology. That is, there exist open sets .A0 , B 0 ⊂ Rn×|P | such that A = C(h¯ ∗ ) ∩ A0 ,

.

B = C(h¯ ∗ ) ∩ B 0

Without loss of generality, we assume .h¯ ∗ ∈ A. According to the way .C(h¯ ∗ ) is constructed, there exists at least one point .h¯ 2 ∈ B such that .h¯ 2 ∈ S(h¯ 1 ) for some ¯ 1 ∈ A. Since .S(h¯ 1 ) is convex, it is connected. We thus consider two nonempty .h . . subsets of .S(h¯ 1 ): .A = S(h¯ 1 ) ∩ A and .B = S(h¯ 1 ) ∩ B. Clearly .A ∩ B = ∅ and

1 ¯ ). In addition, .A ∪ B = S(h A = S(h¯ 1 ) ∩ C(h¯ ∗ ) ∩ A0 = S(h¯ 1 ) ∩ A0

.

B = S(h¯ 1 ) ∩ C(h¯ ∗ ) ∩ B 0 = S(h¯ 1 ) ∩ B 0 which shows that both .A and .B are open in the relative topology. Thus .S(h¯ 1 ) is not connected and we have reached a contradiction.  For a given solution .h¯ ∗ with the (P) property, when the procedure of expanding the solution set described above terminates, one can further take the closure of the set .C(h¯ ∗ ) to include all the boundary points since the limit of any converging sequence of solutions is also a solution. This last statement is due to closedness of the solution set (Proposition 5.6). The entire solution set may have one or more connected components and is generally not convex. In order to provide a global characterization of the solution set, more information on the delay operator, such as generalized monotonicity, is required. As a concluding remark, we point out that the existence results for dynamic user equilibria presented in this chapter are not complete unless the continuity of the effective delay operator is provided, which requires analysis of the arc dynamics and flow propagation in the network. We will discuss dynamic network loading and the continuity of the delay operator in more detail in Chaps. 7 and 8. However, we have presented in this chapter a general framework in which the existence of a DUE can be analyzed, regardless of the network loading model employed. It is significant that ours is the first DUE existence result without the a priori bounding of departure rates (path flows) and the most general constraint relating path flows to the trip

References and Suggested Reading

197

table. In fact, our method of proof successfully overcomes two major hurdles that have stymied other researchers: 1. The sets of feasible controls are intrinsically non-compact in the respective Hilbert spaces; and 2. A direct topological argument requires a priori bounds for the path flows, where those bounds do not arise from any behavioral argument or theory.

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Merchant, D. K., & Numhauser, G. L. (1978a). A model and an algorithm for the dynamic traffic assignment problem. Transportation Science, 12(3), 183–199. Merchant, D. K., & Numhauser, G. L. (1978b). Optimality conditions for a dynamic traffic assignment model. Transportation Science, 12(3), 200–207. Mounce, R. (2006). Convergence in a continuous dynamic queuing model for traffic networks. Transportation Research Part B, 40(9), 779–791. Pang, J., Han, L., Ramadurai, G., & Ukkusuri, S. (2011). A continuous-time linear complementarity system for dynamic user equilibria in single bottleneck traffic flows. Mathematical Programming, Series A, 133(1–2), 437–460. Ran, B., Hall, R. W., & Boyce, D. E. (1996). A link-based variational inequality model for dynamic departure time/route choice. Transportation Research Part B, 30(1), 31–46. Richards, P. I. (1956). Shockwaves on the highway. Operations Research, 4(1), 42–51. Royden, H.L., Fitzpatrick, P. (1988). Real Analysis (Vol. 3). Englewood Cliffs, NJ:: Prentice Hall. Smith, M. J., & Wisten, M. B. (1994). Lyapunov methods for dynamic equilibrium traffic assignment. In Proceedings of the Second Meeting of the EURO Working Group on Urban Traffic and Transportations (pp. 223–245). INRETS: Paris. Smith, M. J., & Wisten, M. B. (1995). A continuous day-to-day traffic assignment model and the existence of a continuous dynamic user equilibrium. Annals of Operations Research, 60(1), 59–79. Vickrey, W. S. (1969). Congestion theory and transport investment. The American Economic Review, 59(2), 251–261. Wie, B., Tobin, R. L., & Carey, M. (2002). The existence, uniqueness and computation of an arcbased dynamic network user equilibrium formulation. Transportation Research Part B, 36(10), 897–918. Yao, T., Friesz, T. L., Wei, M. M., & Yin, Y. (2010). Congestion derivatives for a traffic bottleneck. Transportation Research Part B, 44(10), 1149–1165. Zhu, D. L., & Marcotte, P. (2000). On the existence of solutions to the dynamic user equilibrium problem. Transportation Science, 34(4), 402–414.

Chapter 6

Algorithms for Computing Dynamic User Equilibria

6.1 Fixed-Point Algorithm In order to apply the results developed in Chap. 4 regarding the relationship of dynamic user equilibria to differential variational inequalities (DVI), we must be able to compute the solutions to differential variational inequalities. It should come as no surprise that there is often an equivalent functional fixed-point problem corresponding to a given differential Nash game. This formulation provides an immediate, simple, and sometimes quite effective algorithm for solving the DVI problem. As we shall show later in this section, convergence of this method typically requires Lipschitz continuity and strong monotonicity of the principal operator.

6.1.1 Fixed-Point Algorithm for DUE with Fixed Demand Recall from Sect. 4.2 that a DUE problem and its DVI representation are equivalent to the following fixed-point problem stated in a Hilbert space. Theorem 6.1 (Fixed-point problem equivalent to DUE) Assume that .Ψp (·, h) : [t0 , tf ] → R+ is measurable for all .p ∈ P and .h ∈ Λ1 . Then the fixed-point problem   h∗ = PΛ1 h∗ − αΨ (t, h∗ )

.

(6.1)

is equivalent to the DUE problem, where .PΛ1 [·] is the minimum norm projection onto .Λ1 and .α ∈ R++ . Moreover, . Λ1 =

.

⎧ ⎨ ⎩

h≥0:

dyij = dt



hp (t), yij (t0 ) = 0, yij (tf ) = Qij

p∈Pij

© Springer Nature Switzerland AG 2022 T. L. Friesz, K. Han, Dynamic Network User Equilibrium, Complex Networks and Dynamic Systems 5, https://doi.org/10.1007/978-3-031-25564-9_6

∀(i, j ) ∈ W

⎫ ⎬ ⎭

199

200

6 Algorithms for Computing Dynamic User Equilibria

Under suitable convergence conditions to be discussed later, Theorem 6.1 suggests the iterative scheme  hk+1 = PΛ1 hk − αΨ (t, hk )

.

That is, at the k-th iteration, we need to perform the minimum norm projection by solving the following minimization problem: hk+1 = argmin

.

h

  2 1 k  k − αΨ (h ) − h : h ∈ Λ h  2  1 L 2

(6.2)

which is recognized as a linear-quadratic optimal control problem .



min J k (h) = h

  vij Qij − yij (tf )

(i,j )∈W

+



 

tf t0

(i, j )∈W p∈Pij

2 1 k hp (t) − αΨp (t, h) − hp (t) dt 2

(6.3)

∀ (i, j ) ∈ W.

(6.4)

subject to .

dyij = dt



hp (t)

p∈Pij

yij (t0 ) = 0

∀ (i, j ) ∈ W.

(6.5)

h ≥ 0

(6.6)

Finding dual variables associated with terminal time demand constraints turns out to be relatively easy. Note that the relevant Hamiltonian for (6.3), (6.4), (6.5), and (6.6) is Hk =

.

1 2



 

hkp (t) − αΨp (t, hk ) − hp (t)

(i, j )∈W p∈Pij

2

+

 (i,j )∈W



λij (t)

hp (t)

p∈Pij

(6.7) where each .λij (·) is an adjoint variable obeying .

dλij (t) ∂H k = 0 ∀(i, j ) ∈ W = (−1) dt ∂yij   ∂ vij Qij − yij (t) λij (tf ) = ∀ (i, j ) ∈ W = − vij ∂yij (tf )

6.1 Fixed-Point Algorithm

201

From the above we determine that λij (t) ≡ − vij

.

∀t ∈ [t0 , tf ], (i, j ) ∈ W

The minimum principle implies for any .p ∈ P

 ∂H k = 0 ∂hp     = arg − hkp (t) − αΨp (t, hk ) − hp (t) − vij = 0

hk+1 p (t) = arg

.

Thus we obtain  k+1 .hp (t) = hkp (t) − αΨp (t, hk ) + vij

+

∀ (i, j ) ∈ W, p ∈ Pij

(6.8)

Notice that the following flow conservation constraint applies here.  

tf

.

p∈Pij

t0

hk+1 p (t) dt = Qij

∀ (i, j ) ∈ W

Consequently the dual variable .vij must satisfy  

tf

.

p∈Pij

t0



hkp (t) − αΨp (t, hk ) + vij

+

dt = Qij

∀(i, j ) ∈ W

(6.9)

Recalling that each .vij is time invariant and noticing that the equations of (6.9) are uncoupled among different O-D pairs, we see that a simple root-search algorithm will find the values of .vij satisfying the above conditions (a simple observation of (6.9) suggests that such a .vij exists and is unique). Once the .vij ’s are determined, the new vector of path flows .hk+1 for the next iteration may be computed from (6.8). Remark 6.1 The fixed-point iteration scheme should not be confused with simulation-based approaches since our iterations are merely approximations that are successively refined until convergence produces a realized system state; that is, iterations prior to convergence have no physical meaning. This is in contrast to simulation where each iteration of a simulation model produces a representation of a system state or scenario at a particular point in time. These states are intended to directly map onto the physical world typically with associated performance measures.

202

6 Algorithms for Computing Dynamic User Equilibria

Fixed-Point Algorithm for DUE with Fixed Demand

Step 0. Initialization. Identify an initial feasible solution .h0 ∈ Λ1 and set iteration counter .k = 0 Step 1. Find the effective path delays. Solve the dynamic network loading subproblem with the vector of path departure rates given by .hk , and obtain k .Ψp (t, h ), ∀t ∈ [t0 , tf ], p ∈ P. Step 2. Find the dual variable. For each .(i, j ) ∈ W, solve the following equation for .vij , using root-search algorithms.  

tf

.

p∈Pij

t0



hkp (t) − αΨp (t, hk ) + vij

+

dt = Qij

Step 3. Update the path departure rates. For each .(i, j ) ∈ W and .p ∈ Pij , update the path departure rate as hk+1 p (t) =

.

 hkp (t) − αΨp (t, hk ) + vij

+

∀t ∈ [t0 , tf ]

Step 4. Check for convergence. If  k+1  h − hk L2   . ≤  hk  2 L where . ∈ R++ is a prescribed termination threshold, then terminate the algorithm with output .h∗ ≈ hk . Otherwise, set .k = k + 1 and repeat Step 1 through Step 4.

6.1.2 Fixed-Point Algorithm for DUE with Elastic Demand DUE with elastic demand (E-DUE) is best analyzed in an extended Hilbert space. | P | .  We recall from Sect. 4.4 the product space .E = L2 [t0 , tf ] × R|W | with the

6.1 Fixed-Point Algorithm

203

inner product

.

|P |  . 

X, Y E =

tf

ξi (t) · ηi (t) dt +

t0

i=1

|W | 

uj vj

j =1

 ξ1 (·), . . . , ξ|P | (·), u1 , . . . , u|W | ∈ E   Y = η1 (·), . . . , η|P | (·), v1 , . . . , v|W | ∈ E 

X =

 ⊂ E is defined. The following feasible set .Λ    |W | 2 |P |  .Λ = (h, Q) ∈ (L+ [t0 , tf ]) × R+ : = Qij

∀(i, j ) ∈ W

p∈Pij



tf

hp (t) dt

t0

(6.10)

Note that the elastic demand satisfaction can be trivially re-written as a two-point boundary problem, leading to the following definition of feasible set: 1 = .Λ

|W |

(h, Q) ∈ (L2+ [t0 , tf ])|P | × R+

:

 dyij (t) hp (t), = dt

yij (t0 ) = 0, yij (tf ) = Qij

p∈Pij

∀(i, j ) ∈ W

 (6.11)

Recall from Sect. 4.4 the mapping 1 → E, F :Λ

.

  X = (h, Q) → Ψ (h), Θ − (Q)

We recap the fixed-point formulation of dynamic user equilibrium with elastic demand, originally stated and proven in Sect. 4.4. Theorem 6.2 (Fixed-point formulation of E-DUE) The fixed point problem   X∗ = PΛ1 X∗ − αF(X∗ )

.

(6.12)

is equivalent to the DUE problem with elastic demand, where .α ∈ R++ ; .X∗ = |W | (h∗ , Q∗ ) ∈ (L2+ [t0 , tf ])|P | × R+ ; .PΛ1 [·] is the minimum norm projection onto 1 . .Λ Under suitable convergence conditions to be articulated later, Theorem 6.2 suggests the following iterative computational scheme:  Xk+1 = PΛ1 Xk − αF(Xk )

.

204

6 Algorithms for Computing Dynamic User Equilibria

That is, at the k-th iteration, we need to solve the following minimization problem   k+1 .X = hk+1 , y k+1 (tf )  tf 2 1    k = argmin hp (t) − αΨp (t, hk ) − hp (t) dt t0 2 X (i,j )∈W p∈Pij

+

1 2

 

2   yijk (tf ) − αΘij− y k (tf ) − yij (tf )

(6.13)

(i,j )∈W

subject to .



dyij = dt

∀ (i, j ) ∈ W.

hp (t)

(6.14)

p∈Pij

yij (t0 ) = 0

∀ (i, j ) ∈ W.

(6.15)

h ≥ 0

(6.16)

The Hamiltonian for the above optimal control problem is H =

.

1 2



 

hkp (t) − αΨp (t, hk ) − hp (t)

2

+

(i,j )∈W p∈Pij



λij

(i,j )∈W



hp (t)

p∈Pij

for which the adjoint equations are .

dλij (t) ∂H = 0 = − dt ∂yij

∀(i, j ) ∈ W,

p ∈ Pij ,

t ∈ [t0 , tf ]

(6.17)

2   k (t ) − αΘ − y k (t ) − y (t ) y f ij f (i,j )∈W ij f ij .λij (tf ) = 2 ∂yij (tf )  − k k = − yij (tf ) + αΘij y (tf ) + yij (tf )

(6.18)

and the transversality conditions read, for all .(i, j ) ∈ W: 1∂





By the minimum principle, we enforce the following minimization problem .

min H h

s.t.

−h ≤ 0

for which the Kuhn-Tucker conditions are (we denote the K-T point by .hk+1 ) .

 − hkp (t) − αΨp (t, hk ) − hk+1 p (t) + λij = ρp

∀(i, j ) ∈ W,

p ∈ Pij . (6.19)

6.1 Fixed-Point Algorithm

205

ρp hk+1 = 0 p

∀(i, j ) ∈ W,

p ∈ Pij . (6.20)

ρp ≥ 0

∀(i, j ) ∈ W,

p ∈ Pij (6.21)

Thus the optimality conditions for system (6.13)–(6.16) are: λij (t) ≡ − yijk (tf ) + αΘij− [y k (tf )] + yijk+1 (tf )

.

0 ≤

.

∀(i, j ) ∈ W

(6.22)



∂H k = arg =0 ∂hp

hk+1 p (t)

    = arg − hkp (t) − αΨp (t, hk ) − hp (t) + λij = 0    = arg − hkp (t) − αΨp (t, hk ) − hp (t) − yijk (tf ) + αΘij− [y k (tf )]  (6.23) + yijk+1 (tf ) = 0 In other words, given .(i, j ) ∈ W and .p ∈ Pij , .hk+1 p (·) is determined as 



hkp − αΨp (t, hk ) + yijk (tf ) − αΘij− [y k (tf )] − yijk+1 (tf )  = hkp (t) − αΨp (t, hk ) + Qkij + αΘij [Qk ] − Qk+1 ij

hk+1 p (t) =

.

+

(6.24)

+

where k .Qij

≡

yijk (tf )

=

  p∈Pij

tf t0

hkp (t) dt

∀(i, j ) ∈ W

. and .[x]+ = max{0, x} for .x ∈ R. Notice that for all .(i, j ) ∈ W, .Qk+1 ij must satisfy Qk+1 = ij

 

tf

.

p∈Pij

=

t0

 

p∈Pij

hk+1 p (t)dt

tf t0



  dt hkp (t) − αΨp (t, hk ) + Qkij + αΘij Qk − Qk+1 ij +

(6.25)

206

6 Algorithms for Computing Dynamic User Equilibria

Finally, we will show the existence and uniqueness of .Qk+1 ∈ R+ that ij satisfies (6.25). We slightly re-write (6.25) as  



  dt − Qk+1 = 0 hkp (t) − αΨp (t, hk ) + Qkij + αΘij Qk − Qk+1 ij ij

tf

.

p∈Pij

+

t0

(6.26) Two cases arise: (i) .hkp (t) − αΨp (t, hk ) + Qkij + αΘij [Qk ] ≤ 0 .∀v (t) ∈ [t0 , tf ], ∀p ∈ Pij . Then clearly .Qk+1 = 0 is the only solution to (6.26) ij (ii) .hkp (t) − αΨp (t, hk ) + Qkij + αΘij [Qk ] > 0 for some .p ∈ Pij and for .t ∈ B ⊂ [t0 , tf ] where .B is a set with positive measure. We call the left hand side k+1 of (6.26) .f (Qk+1 ij ), which is a continuous function of .Qij . According to the hypothesis, the following hold f (0) > 0,

.

f (Qk+1 ij ) < 0

if Qk+1 ij is very large

Therefore, by the Intermediate Value Theorem, there must exist at least one such that .f (Qk+1 value of .Qk+1 ij ij ) vanishes. The uniqueness of such a solution follows by observing that .f (·), as a function of .Qk+1 ij , is strictly decreasing. k+1 k+1 k+1 Therefore, .X = (h , Q ) given by (6.24) and (6.25) is unique. This suggests that one can conduct a simple root search procedure to find the unique value of .Qk+1 ij that satisfies (6.25). With the preceding discussion, we can now state the algorithm for computing DUE with elastic demand. Fixed-Point Algorithm for DUE with Elastic Demand

 and Step 0. Initialization. Identify an initial feasible solution .(h0 , Q0 ) ∈ Λ set iteration counter .k = 0 Step 1. Find the effective path delays. Solve the dynamic network loading problem with path departure rates given by .hk , and obtain the effective path delays .Ψp (t, hk ), ∀t ∈ [t0 , tf ], p ∈ P. Step 2. Update trip matrix. For each .(i, j ) ∈ W, solve the following equation for .Qk+1 ij , using root-search algorithms.  

tf

.

p∈Pij

t0



  dt −Qk+1 = 0 hkp (t)−αΨp (t, hk )+Qkij +αΘij Qk −Qk+1 ij ij +

(continued)

6.1 Fixed-Point Algorithm

207

Step 3. Update path departure rates. For each .(i, j ) ∈ W and .p ∈ Pij , update each path departure rate as hk+1 p (t) =

.



hkp (t) − αΨp (t, hk ) + Qkij + αΘij [Qk ] − Qk+1 ij

+

∀t ∈ [t0 , tf ] Step 4. Check for convergence. If

.

 k+1  (h , Qk+1 ) − (hk , Qk ) E (hk , Qk )E

≤

where . ∈ R++ is a prescribed termination threshold, then terminate the algorithm with output .X∗ ≈ (hk , Qk ). Otherwise, set .k = k + 1 and repeat Step 1 through Step 4.

6.1.3 Fixed-Point Algorithm for DUE with Bounded Rationality Let us recall the fixed-point formulation of boundedly rational dynamic user equilibrium with variable tolerance (VT-BR-DUE). Notice that the computational algorithms discussed in this section apply equally to boundedly rational dynamic user equilibrium with fixed and exogenous tolerances (BR-DUE). The new operator associated with VT-BR-DUE is recalled:  | P | Φ ε : Λ1 → L2+ [t0 , tf ] ,

.

  h → Φpε (·, h) : p ∈ P

(6.27)

where ε .Φp (t,



h) = max Ψp (t, h),

p vij (h) + εij (h)



 −

p εij (h) −

 min

q∈Pij

q εij (h)



(6.28) Theorem 6.3 (Fixed-point formulation of VT-BR-DUE) Assume that .Φ ε (·, h) defined in (6.27) and (6.28) is measurable and strictly positive for all .p ∈ P and .h ∈ Λ1 . Then the fixed-point problem   h∗ = PΛ1 h∗ − αΦ ε (t, h∗ )

.

(6.29)

208

6 Algorithms for Computing Dynamic User Equilibria

is equivalent to the VT-BR-DUE problem, where .PΛ1 [·] is the minimum norm projection onto .Λ1 and .α > 0 is a fixed constant. Moreover, . .Λ1 =

⎧ ⎨

 dyij hp (t), yij (t0 ) = 0, yij (tf ) = Qij = h≥0: ⎩ dt

∀(i, j ) ∈ W

p∈Pij

⎫ ⎬ ⎭

Under suitable convergence conditions to be discussed later, Theorem 6.3 suggests the iterative scheme  hk+1 = PΛ1 hk − αΦ ε (t, hk )

.

That is, at the k-th iteration, we need to perform the minimum norm projection by solving the following minimization problem: hk+1 = argmin

.

h

  2 1 k  h − αΦ ε (hk ) − h 2 : h ∈ Λ1 L 2

which is recognized as a linear-quadratic optimal control problem .



min J k (h) = h

  vij Qij − yij (tf )

(i,j )∈W

+

 



(i, j )∈W p∈Pij

tf t0

2 1 k hp (t) − αΦpε (t, h) − hp (t) dt 2

(6.30)

subject to .

dyij = dt



hp (t)

∀ (i, j ) ∈ W.

(6.31)

p∈Pij

yij (t0 ) = 0

∀ (i, j ) ∈ W.

(6.32)

h ≥ 0

(6.33)

Finding dual variables associated with terminal time demand constraints turns out to be relatively easy. Note that the relevant Hamiltonian for (6.30), (6.31), (6.32), and (6.33) is Hk =

.

1 2



 

(i, j )∈W p∈Pij

hkp (t) − αΦpε (t, hk ) − hp (t)

2

+

 (i,j )∈W

λij (t)



hp (t)

p∈Pij

(6.34)

6.1 Fixed-Point Algorithm

209

where each .λij (·) is an adjoint variable obeying .

dλij (t) ∂H k = 0 ∀(i, j ) ∈ W = (−1) dt ∂yij   ∂ vij Qij − yij (t) λij (tf ) = ∀ (i, j ) ∈ W = − vij ∂yij (tf )

From the above we determine that λij (t) ≡ − vij

.

∀t ∈ [t0 , tf ], (i, j ) ∈ W

The minimum principle implies for any .p ∈ P,

 ∂H k = 0 ∂hp     = arg − hkp (t) − αΦpε (t, hk ) − hp (t) − vij = 0

hk+1 p (t) = arg

.

Thus we obtain  k+1 .hp (t) = hkp (t) − αΦpε (t, hk ) + vij

+

∀ (i, j ) ∈ W, p ∈ Pij

(6.35)

Notice that the following flow conservation constraint applies here.  

tf

.

p∈Pij

t0

hk+1 p (t) dt = Qij

∀ (i, j ) ∈ W

Consequently the dual variable .vij must satisfy  

tf

.

p∈Pij

t0



hkp (t) − αΦpε (t, hk ) + vij

+

dt = Qij

∀(i, j ) ∈ W

(6.36)

Recalling that each .vij is time invariant and noticing that the equations of (6.36) are uncoupled among different O-D pairs, we see that a simple root-search algorithm will find the values of .vij satisfying the above conditions (a simple observation of (6.36) suggests that such a .vij exists and is unique). Once the .vij ’s are determined, the new vector of path flows .hk+1 for the next iteration may be computed from (6.35). The following fixed-point algorithm is for VT-BR-DUE problems. However, it also applies to BR-DUE with fixed tolerances by replacing .Φ ε with .φ ε , which is defined in (4.101) and (4.102).

210

6 Algorithms for Computing Dynamic User Equilibria

Fixed-Point Algorithm for DUE with Bounded Rationality

Step 0. Initialization. Identify an initial feasible solution .h0 ∈ Λ. Set the iteration counter .k = 0. Step 1. Evaluate the principal operator Solve the dynamic network loading problem with path departure rates given by .hk , and obtain the effective path delays .Ψp (·, hk ), ∀p ∈ P. Let .vij (hk ) be the minimum effective delay within O-D pair .(i, j ). Then compute:          q p p Φpε t, hk = max Ψp t, hk , vij (hk ) + εij (hk ) − εij (hk )− min εij (hk )

.

q∈Pij

for all .p ∈ Pij , .t ∈ [t0 , tf ]. Step 2. Update path departure rates. For each .(i, j ) ∈ W, solve the . following equation for .μij , using root-search algorithms (here .[x]+ = max{0, x}).  

tf

.

p∈Pij

t0



  hkp (t) − αΦpε t, hk + μij dt = Qij +

Then update the next iterate .hk+1 = {hk+1 : p ∈ P} where p hk+1 p (t) =

.



  hkp (t) − αΦpε t, hk + μij

+

∀t ∈ [t0 , tf ], p ∈ Pij , (i, j ) ∈ W Step 3. Check for convergence. Terminate the algorithm with output .h∗ ≈ hk if      k+1    . h − hk  2 hk  2 ≤  L

L

where . ∈ R++ is a prescribed termination threshold. Otherwise, set .k = k+1 and repeat Step 1 through Step 3.

6.2 Self-adaptive Projection Algorithm

211

6.2 Self-adaptive Projection Algorithm The second type of algorithm we consider is the so-called self-adaptive projection method. This method is originally proposed by Han and Lo (2002) for solving generic variational inequalities. The applicability of this algorithm for solving DUE, E-DUE, and VT-BR-DUE is made possible by the variational inequality formulations that we have established in Chap. 3 for these types of DUEs. As we shall see later, this method relies on the pseudo monotonicity of the delay operator for convergence to hold.

6.2.1 Self-adaptive Projection Algorithm for DUE with Fixed Demand We recall from Sect. 3.2.2 the variational inequality (VI) formulation of DUE with fixed demand: Theorem 6.4 (VI formulation of DUE with fixed demand) The simultaneous route-and-departure-time dynamic user equilibrium as in Definition 3.1 is equivalent to the following variational inequality problem in the Hilbert space  2 | P | . L [t0 , tf ]  .

Ψ (h∗ ), h − h∗



≥ 0

∀h ∈ Λ

(6.37)

where the inner product in the Hilbert space is defined as  .

Ψ (h∗ ), h − h∗

  .  = p∈P

tf

t0

Ψp (t, h∗ )[hp (t) − h∗p (t)] dt

and the feasible set .Λ is given by Λ =

.

⎧ ⎨

h≥0:

⎩

  p∈Pij

tf

hp (t) dt = Qij

t0

∀ (i, j ) ∈ W

⎫ ⎬ ⎭

  |P | ⊆ L2+ t0 , tf

In order to illustrate the self-adaptive projection method for solving (6.37), we begin with some basic notations. As before, .PΛ [·] denotes the projection onto the set .Λ. Define the residual   . r(h; β) = h − PΛ h − β Ψ (h)

.

h ∈ Λ, β > 0

(6.38)

212

6 Algorithms for Computing Dynamic User Equilibria

where the projection .h¯ = PΛ [h − βΨ (h)] is explicitly given via   ⎧ ∀t ∈ [t0 , tf ], ∀p ∈ Pij , ∀(i, j ) ∈ W ⎪ h¯ p (t) = hp (t) − βΨp (t, h) + vij + ⎨  tf   . hp (t) − βΨp (t, h) + vij + dt = Qij ∀(i j ) ∈ W ⎪ ⎩ p∈Pij

t0

Notice that the residual is zero if and only if h is a solution of the VI. Given .α, β > 0, let   . d(h; α, β) = αr(h; β) + βΨ h − αr(h; β) .    . g(h; α, β) = α r(h; β) − β Ψ (h) − Ψ (h − αr(h; β)) .

.

. r(h; β) , g(h; α, β) ρ(h; α, β) = d(h; α, β)2L2

(6.39) (6.40) (6.41)

Self-adaptive Projection Algorithm for DUE with Fixed Demand

Step 0. Initialization. Choose fixed parameters .μ ∈ (0, 1), γ ∈ (0, 2), θ > 1, and .L ∈ (0, 1). Let . > 0 be the termination threshold. Identify an initial feasible solution .h0 = (h0 , Q0 ) ∈ Λ, and set iteration counter .k = 0. Let .αk = 1. Step 1. Compute the residual Set .βk = min{1, θ αk }. Perform dynamic network loading, and compute the residual .r(hk ; βk ) according to (6.38). If .

   k  r(h ; βk )

   k / h  2

L2

L

≤

terminate the algorithm; otherwise, continue to Step 2. Step 2. Find the smallest nonnegative integer .mk such that .αk+1 = βk μmk satisfies     k    k   k k .βk Ψ (h ) − Ψ h − αk+1 r(h ; βk )  ≤ L ; β ) (6.42) r(h  2 k 2 L

L

Step 3. Update departure rates. Compute   hk+1 = PΛ hk − γρ(hk ; αk+1 , βk )d(hk ; αk+1 , βk )

.

Set .k = k + 1 and go to Step 1.

(6.43)

6.2 Self-adaptive Projection Algorithm

213

Equation (6.42) requires evaluation of .Ψ at the point .hk − αk+1 r(hk ; βk ). We show here that such a point always belongs to .Λ, the domain of .Ψ . Notice that k k k k .r(h ; βk ) = h − PΛ [h − βk Ψ (h )]; thus hk − αk+1 r(hk ; βk ) = (1 − αk+1 )hk + αk+1 PΛ [hk − βk Ψ (hk )] ∈ Λ

.

since both .hk and .PΛ [hk − βk Ψ (hk )] belong to the convex set .Λ. In Step 2 of the above algorithm, one is required to test a range of integers, starting from zero, in order to find the smallest integer .mk . We show below that such a procedure can always terminate within a finite number of trials. Assume that 1 .Ψ is a continuous operator, and observe that .αk+1 → 0 as .mk → +∞. There exists .N > 0 such that, for every .mk > N , we have     L r(hk ; βk )L2 L   k k k . Ψ (h ) − Ψ (h − αk+1 r(h ; βk )) ≤ ≤ , L2 βk βk which is (6.42). In case .mk > 1, the algorithm requires more than one evaluation of the operator (i.e., more than one dynamic network loading procedures) within an iteration, which is less efficient than the projection algorithm. However, compared to the fixed-point algorithm, the self-adaptive projection algorithm entails relaxed monotonicity of the operator .Ψ to ensure convergence.

6.2.2 Self-adaptive Projection Algorithm for DUE with Elastic Demand We recall from Sect. 3.3.2 the variational inequality (VI) formulation of DUE with elastic demand: Theorem 6.5 (DUE with elastic demand equivalent to a variational inequality problem) Assume .Ψp (·, h) : [t0 , tf ] → R++ is measurable and strictly positive  Also assume that the elastic travel for all .p ∈ P and all h such that .(h, Q) ∈ Λ. demand function is invertible with inverse .Θij (·) for all .(i, j ) ∈ W. Then a pair ∗ ∗  is a DUE with elastic demand as in Definition 3.2 if and only it .(h , Q ) ∈ Λ solves the following variational inequality:  such that find (h∗ , Q∗ ) ∈ Λ  tf Ψp (t, h∗ )(hp − h∗p )dt .

−



1 Continuity

p∈P

t0

(i, j )∈W

Θij



Q∗



Qij − Q∗ij

 ∀(h, Q) ∈ Λ



⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ≥ 0⎪ ⎪ ⎪ ⎪ ⎭

  V I Ψ, Θ, [t0 , tf ]

(6.44)

of the operator .Ψ is stated as one of the convergence conditions for this method.

214

6 Algorithms for Computing Dynamic User Equilibria

where  |W |  = (h, Q) ∈ (L2+ [t0 , tf ])|P | × R+ .Λ : ∀(i, j ) ∈ W

= Qij



  p∈Pij

tf

hp (t) dt t0

By introducing more compact notations: .  X = (h, Q) ∈ Λ,

.

   → E, X → Ψ (h), Θ − [Q] F :Λ

  V I Ψ, Θ, [t0 , tf ] can be written as

.

 .

F(X∗ ), X − X∗

 E

≥ 0

 ∀X ∈ Λ

(6.45)

We begin with some basic notations needed to articulate the self-adaptive  Define projection method. As before, .PΛ[·] denotes the projection onto the set .Λ. the residual   . r(X; β) = X − PΛ X − β F(X)

.

 β > 0 X ∈ Λ,

(6.46)

. ¯ ¯ Q) = PΛ[X − βF(X)] is explicitly given via where the projection .X¯ = (h,   ⎧ ¯ ¯ ⎪ ⎨ hp (t) =t hp (t) − βΨp (t, h) +Qij + βΘij [Q] − Qij + ∀t ∈ [t0 , tf ], p ∈ Pij  f   . ¯ ij dt = Q ¯ ij hp (t) − βΨp (t, h) + Qij + βΘij [Q] − Q ⎪ + ⎩ t0 p∈Pij

for all .(i, j ) ∈ W. Notice that the residual is zero if and only if X is a solution of the VI. Given .α, β > 0, let   . d(X; α, β) = αr(X; β) + βF X − αr(X; β) .    . g(X; α, β) = α r(X; β) − β F(X) − F(X − αr(X; β)) .

.

. r(X; β) , g(X; α, β)E ρ(X; α, β) = d(X; α, β)2E

(6.47) (6.48) (6.49)

6.2 Self-adaptive Projection Algorithm

215

Self-adaptive Projection Algorithm for DUE with Elastic Demand

Step 0. Initialization. Choose fixed parameters .μ ∈ (0, 1), γ ∈ (0, 2), θ > 1, and .L ∈ (0, 1). Let . > 0 be the termination threshold. Identify an initial  and set iteration counter .k = 0. Let feasible solution .X0 = (h0 , Q0 ) ∈ Λ, .αk = 1. Step 1. Compute the residual Set .βk = min{1, θ αk }. Perform dynamic k network loading,   compute the residual .r(X ; βk ) according to (6.46). If   and k k     . r(X ; βk ) / X E ≤ , terminate the algorithm; otherwise, continue to E Step 2. Step 2. Find the smallest nonnegative integer .mk such that .αk+1 = βk μmk satisfies     k      k k .βk F(X ) − F X − αk+1 r(X ; βk )  ≤ L r(Xk ; βk ) (6.50) E

E

Step 3. Update departure rates. Compute   Xk+1 = PΛ Xk − γρ(Xk ; αk+1 , βk )d(Xk ; αk+1 , βk )

.

(6.51)

Set .k = k + 1 and go to Step 1.

Equation (6.50) requires evaluation of .F at the point .Xk − αk+1 r(Xk ; βk ). We  the domain of .F. Notice that here show that such a point always belongs to .Λ, k k k − β F(X k )]; thus .r(X ; βk ) = X − PΛ [X  k  Xk − αk+1 r(Xk ; βk ) = (1 − αk+1 )Xk + αk+1 PΛ[Xk − βk F(X k )] ∈ Λ

.

 since both .Xk and .PΛ[Xk − βk F(X k )] belong to the convex set .Λ. In Step 2 of the above algorithm, one is required to test a range of integers, starting from zero, in order to find the smallest integer .mk . We show below that such a procedure can always terminate within finite number of trials. Assume that 2 .F is a continuous operator and observe that .αk+1 → 0 as .mk → +∞. There exists .N > 0 such that for every .mk > N , we have     k k k . F(X ) − F(X − αk+1 r(X ; βk ))

2 Continuity

E

  L r(Xk ; βk )E L ≤ ≤ βk βk

of the operator .F is stated as one of the convergence conditions for this method.

216

6 Algorithms for Computing Dynamic User Equilibria

which is (6.50). In case .mk > 1, the algorithm requires more than one evaluation of the operator (i.e., more than one dynamic network loading procedure) within an iteration.

6.2.3 Self-adaptive Projection Algorithm for DUE with Bounded Rationality We recall from Sect. 3.4.2 the VI formulation of boundedly rational dynamic user equilibrium with variable tolerance (VT-BR-DUE). Notice that the computational algorithm discussed here applies equally to boundedly rational dynamic user equilibrium with fixed and exogenous tolerances (BR-DUE). The new operator associated with VT-BR-DUE is recalled:  | P | Φ ε : Λ → L2+ [t0 , tf ] ,

  h → Φpε (·, h) : p ∈ P

.

(6.52)

where      p p q Φpε (t, h) = max Ψp (t, h), vij (h) + εij (h) − εij (h) − min εij (h)

.

q∈Pij

(6.53) Theorem 6.6 (VT-BR-DUE equivalent to a variational inequality) Given .ε(·) : |P | Λ → R+ , define .Φ ε (·) according to (6.52) and (6.53). Then, a vector of path departure rates .h∗ ∈ Λ is a VT-BR-DUE solution if and only if it solves the following variational inequality.  .

Φ ε (h∗ ), h − h∗



≥ 0

∀h ∈ Λ

(6.54)

where Λ =

.

⎧ ⎨ ⎩

h≥0:

  p∈Pij

tf

hp (t) dt = Qij ∀ (i, j ) ∈ W

t0

⎫ ⎬ ⎭

  |P | ⊆ L2+ t0 , tf

In order to illustrate the self-adaptive projection method for solving (6.54), we begin with some basic notations. As before, .PΛ [·] denotes the projection onto the set .Λ. Define the residual   . r(h; β) = h − PΛ h − β Φ ε (h)

.

h ∈ Λ, β > 0

(6.55)

6.2 Self-adaptive Projection Algorithm

217

where the projection .h¯ = PΛ [h − βΦ ε (h)] is explicitly given via  ⎧ ⎪ h¯ p (t) = hp (t) − βΦpε (t, h) + vij ∀t ∈ [t0 , tf ], ∀p ∈ Pij , ∀(i, j ) ∈ W ⎪ ⎨ +  tf   . ∀(i j ) ∈ W hp (t) − βΦpε (t, h) + vij dt = Qij ⎪ ⎪ ⎩ + p∈Pij

t0

Notice that the residual is zero if and only if h is a solution of the VI. Given .α, β > 0, let   . d(h; α, β) = αr(h; β) + βΦ ε h − αr(h; β) .    . g(h; α, β) = α r(h; β) − β Φ ε (h) − Φ ε (h − αr(h; β)) .

.

. r(h; β) , g(h; α, β) ρ(h; α, β) = d(h; α, β)2L2

(6.56) (6.57) (6.58)

Self-adaptive Projection Algorithm for DUE with Bounded Rationality

Step 0. Initialization. Choose fixed parameters .μ ∈ (0, 1), γ ∈ (0, 2), θ > 1, and .L ∈ (0, 1). Let . > 0 be the termination threshold. Identify an initial feasible solution .h0 = (h0 , Q0 ) ∈ Λ, and set iteration counter .k = 0. Let .αk = 1. Step 1. Compute the residual Set .βk = min{1, θ αk }. Perform dynamic network loading, and compute the residual .r(hk ; βk ) according to (6.55). If .

    k r(h ; βk )

   k / h  2

L2

L

≤

terminate the algorithm; otherwise, continue to Step 2. Step 2. Find the smallest nonnegative integer .mk such that .αk+1 = βk μmk satisfies       ε k   k  ε k k .βk Φ (h ) − Φ h − αk+1 r(h ; βk )  ≤ L ; β ) (6.59) r(h  2 k 2 L

L

Step 3. Update departure rates. Compute   hk+1 = PΛ hk − γρ(hk ; αk+1 , βk )d(hk ; αk+1 , βk )

.

Set .k = k + 1 and go to Step 1.

(6.60)

218

6 Algorithms for Computing Dynamic User Equilibria

Equation (6.59) requires evaluation of .Φ ε at the point .hk − αk+1 r(hk ; βk ). We show here that such a point always belongs to .Λ, the domain of .Φ ε . Notice that k k k ε k .r(h ; βk ) = h − PΛ [h − βk Φ (h )]; thus hk − αk+1 r(hk ; βk ) = (1 − αk+1 )hk + αk+1 PΛ [hk − βk Φ ε (hk )] ∈ Λ

.

since both .hk and .PΛ [hk − βk Φ ε (hk )] belong to the convex set .Λ. In Step 2 of the above algorithm, one is required to test a range of integers, starting from zero, in order to find the smallest integer .mk . We show below that such a procedure can always terminate within finite number of trials. Assume that ε 3 .Φ is a continuous operator and observe that .αk+1 → 0 as .mk → +∞. There exists .N > 0 such that, for every .mk > N , we have     ε k ε k k . Φ (h ) − Φ (h − αk+1 r(h ; βk ))

L2

  L r(hk ; βk )L2 L ≤ ≤ , βk βk

which is (6.59). In case .mk > 1, the algorithm requires more than one evaluation of the operator (i.e., more than one dynamic network loading procedure) within an iteration, which is less efficient than the projection algorithm.

6.3 Proximal Point Method The proximal point method (PPM) (Konnov, 2003) is a popular method for solving optimization problems and variational inequalities. It replaces the original problem with a sequence of regularized problems, each of which can be solved with standard algorithms due to improved regularity. The PPM is known to converge in the prescence of appropriate generalized monotonicity (Allevi et al., 2006). In this paper we apply the PPM to solve E-DUE problems with some even more relaxed conditions for convergence than previous studies. The proximal point method has been further developed in this book for solving a range of DUE problems.

6.3.1 Proximal Point Method for DUE with Fixed Demand Let us consider the VI formulation of DUE with fixed demand (Theorem 6.4):  .

Ψ (h∗ ), h − h∗



≥ 0

∀h ∈ Λ

The proximal point method for solving this VI is summarized below.

3 Continuity

of the operator .Φ ε is stated as one of the convergence conditions for this method.

6.3 Proximal Point Method

219

Proximal Point Method for DUE with Fixed Demand

Step 0. Initialization. Identify an initial feasible solution .h0 ∈ Λ. Fix a large constant .a > 0 and set a tolerance parameter .δ > 0. Set the iteration counter .k = 0. Step 1. Regularized VI Solve the following variational inequality for .hk+1 ∈ Λ:  .

 Ψ (hk+1 ) + a(hk+1 − hk ) , h − hk+1 ≥ 0

∀h ∈ Λ

(6.61)

Step 2. Check for convergence. Terminate the algorithm if .hk+1 − hk L2 ≤ δ aD , where D is the diameter of the set .Λ. Otherwise, set .k = k + 1 and repeat Step 1 through Step 2.

The key step of the proximal point method (PPM) is to solve the VI (6.61), which enjoys a significantly improved regularity than the original VI problem. To see this, we rewrite Ψ (hk+1 ) + a(hk+1 − hk )

.

as (Ψ + aI )(hk+1 ) − ahk

.

where I is the identity map. If .Ψ is weakly monotone with constant .−K where K > 0, which is a mild condition that is satisfied by a wide class of operators, then k+1 ) − ahk is a strongly monotone operator acting on .hk+1 provided .(Ψ + aI )(h that .a > K. Thus, by choosing a large enough, the VI (6.61) can be solved by any existing algorithm with satisfactory convergence result. For example, one can apply the fixed-point algorithm to solve the intermediate VI: Find .hk+1 such that ! " k+1 . (Ψ + aI )(h ) − ahk , h − hk+1 ≥ 0 ∀h ∈ Λ (6.62) .

We slightly modify the fixed-point algorithm presented in Sect. 6.1.1 by replacing .Ψ with the operator .Ψ +aI −ahk , which enjoys an improved regularity. The algorithm for solving the intermediate, regularized, VI is shown below.

220

6 Algorithms for Computing Dynamic User Equilibria

Solving the Intermediate VI (6.62) in the PPM

Input. Fix a large number .a > 0 and a positive constant .α > 0. The k-th iterate from the PPM, .hk ∈ Λ, is known. Step 0. Initialization. Identify an initial feasible solution .h0 ∈ Λ, and set iteration counter .n = 0. Step 1. Find the effective path delays. Solve the dynamic network loading subproblem with the vector of path departure rates given by .hn , and obtain n .Ψp (t, h ), ∀t ∈ [t0 , tf ], p ∈ P. Step 2. Find the dual variable. For each .(i, j ) ∈ W, solve the following equation for .vij , using root-search algorithms:  

tf

.

p∈Pij

t0

  hnp (t) − α Ψp (t, hn ) + ahn − ahk + vij dt = Qij



+

Step 3. Update the path departure rates. For each .(i, j ) ∈ W and .p ∈ Pij , update the path departure rate as    n+1 .hp (t) = hnp (t) − α Ψp (t, hn ) + ahn − ahk + vij ∀t ∈ [t0 , tf ] +

Step 4. Check for convergence. If  n+1  h − hn L2   . ≤  hn  2 L

where . ∈ R++ is a prescribed termination threshold, and then terminate the algorithm with output .hk+1 ≈ hn . Otherwise, set .n = n + 1 and repeat Step 1 through Step 4.

6.3.2 Proximal Point Method for DUE with Elastic Demand We recall from Theorem 6.5 the VI formulation of E-DUE in the extended Hilbert space:  .

F(X∗ ), X − X∗

 E

≥ 0

 ∀X ∈ Λ,

6.3 Proximal Point Method

221

 and where .X = (h, Q) ∈ Λ  |W |  = (h, Q) ∈ (L2+ [t0 , tf ])|P | × R+ .Λ : = Qij

 ∀(i, j ) ∈ W ,

  p∈Pij

tf

hp (t) dt t0

   → E, .X → Ψ (h), Θ − [Q] . while .F : Λ Given the generic VI representation of the E-DUE problem, we next present the proximal point method for its solution. Proximal Point Method for DUE with Elastic Demand

Step 0. Initialization. Identify an initial feasible solution .X0 = (h0 , Q0 ) ∈  Fix a large constant .a > 0 and set a tolerance parameter .δ > 0. Set the Λ. iteration counter .k = 0. Step 1. Regularized VI Solve the following variational inequality for .Xk+1 = (hk+1 , Qk+1 ):  .

F(Xk+1 ) + a(Xk+1 − Xk ) , X − Xk+1

 E

≥ 0

 ∀X ∈ Λ

(6.63)

Step 2. Check for convergence. Terminate the algorithm if .Xk+1 −Xk E ≤ δ  aD , where D is the diameter of the set .Λ. Otherwise, set .k = k + 1 and repeat Step 1 through Step 2.

Again, if .F is weakly monotone with constant .−K where .K > 0, then the principal operator .(F + aI )(X k+1 ) − aX k acting on .Xk+1 is strongly monotone  such that provided that .a > K. And thus for the intermediate VI, find .Xk+1 ∈ Λ  .

  F + aI (Xk+1 ) − aXk , X − Xk+1 E ≥ 0

 ∀X ∈ Λ

(6.64)

can be solved using any standard method that converges for strong monotonicity. Again, as an example, we write down the fixed-point algorithm for solving this intermediate VI.

222

6 Algorithms for Computing Dynamic User Equilibria

Solving the Intermediate VI (6.64) in the PPM

Input. Fix a large number .a > 0 and a positive constant .α > 0. The k-th  is known. iterate from the PPM, .Xk = (hk , Qk ) ∈ Λ,  and Step 0. Initialization. Identify an initial feasible solution .(h0 , Q0 ) ∈ Λ, set iteration counter .n = 0. Step 1. Find the effective path delays. Solve the dynamic network loading problem with path departure rates given by .hn , and obtain the effective path delays .Ψp (t, hn ), ∀t ∈ [t0 , tf ], p ∈ P. Step 2. Update trip matrix. For each .(i, j ) ∈ W, solve the following equation for .Qn+1 ij , using root-search algorithms.  

tf

.

p∈Pij

t0



  hnp (t) − α Ψp (t, hn ) + ahnp (t) − ahkp (t)

    dt = Qn+1 + Qnij − α Θij Qn + aQnij − aQkij − Qn+1 ij ij +

(6.65)

Step 3. Update path departure rates. For each .(i, j ) ∈ W and .p ∈ Pij , update each path departure rate as    n n n k hn+1 p (t) = hp (t) − α Ψp (t, h ) + ahp (t) − ahp (t)     + Qnij − α Θij Qn + aQnij − aQkij − Qn+1 ij

.

+

∀t ∈ [t0 , tf ] (6.66)

Step 4. Check for convergence. If

.

 n+1  (h , Qn+1 ) − (hn , Qn ) E (hn , Qn )E

≤ 

where . ∈ R++ is a prescribed termination threshold, then terminate the algorithm with output .Xk+1 ≈ (hn , Qn ). Otherwise, set .n = n + 1 and repeat Step 1 through Step 4.

6.3 Proximal Point Method

223

6.3.3 Proximal Point Method for DUE with Bounded Rationality The variational inequality for the VT-BR-DUE problem, established in Sect. 3.4.2, is recapped as follows. Here, we consider themost general case where the vector of  p tolerances is endogenously determined: .ε = εij (·) : p ∈ Pij , (i, j ) ∈ W : Λ → |W |

R+ . The principal operator of the VI is given as  | P | Φ ε : Λ → L2+ [t0 , tf ] ,

  h → Φpε (·, h) : p ∈ P

.

(6.67)

where   p Φpε (t, h) = max Ψp (t, h), vij (h) + εij (h)    p q ∀p ∈ Pij − εij (h) − min εij (h)

.

q∈Pij

(6.68)

Theorem 6.7 (VT-BR-DUE equivalent to a variational inequality) Given .ε(·) : |P | Λ → R+ , define .Φ ε (·) according to (6.67) and (6.68). Then, a vector of path departure rates .h∗ ∈ Λ is a VT-BR-DUE solution if and only if it solves the following variational inequality: Find .h∗ ∈ Λ, such that  .

Φ ε (h∗ ), h − h∗



≥ 0

∀h ∈ Λ

(6.69)

The proximal point method for solving this VI is summarized below. Proximal Point Method for DUE with Fixed Demand

Step 0. Initialization. Identify an initial feasible solution .h0 ∈ Λ. Fix a large constant .a > 0 and set a tolerance parameter .δ > 0. Set the iteration counter .k = 0. Step 1. Regularized VI Solve the following variational inequality for .hk+1 ∈ Λ:  .

 Φ ε (hk+1 ) + a(hk+1 − hk ) , h − hk+1 ≥ 0

∀h ∈ Λ

(6.70)

Step 2. Check for convergence. Terminate the algorithm if .hk+1 − hk L2 ≤ δ aD , where D is the diameter of the set .Λ. Otherwise, set .k = k + 1 and repeat Step 1 through Step 2.

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6 Algorithms for Computing Dynamic User Equilibria

Notice that the intermediate VI in the proximal point method  .

 Φ ε (hk+1 ) + a(hk+1 − hk ) , h − hk+1 ≥ 0

∀h ∈ Λ

has the principal operator .(Φ ε + aI )(hk+1 ) − ahk If .Φ ε is weakly monotone with constant .−K where .K > 0, then .(Φ ε + aI )(hk+1 ) − ahk is a strongly monotone operator acting on .hk+1 provided that .a > K. Thus, by choosing a large enough, the VI (6.70) can be solved by any existing algorithm with satisfactory convergence result. For example, one can apply the fixed-point algorithm to solve the intermediate VI: Find .hk+1 such that ! " ε k+1 . (Φ + aI )(h ) − ahk , h − hk+1 ≥ 0 ∀h ∈ Λ (6.71) We slightly modify the fixed-point algorithm presented in Sect. 6.1.3 by replacing Φ ε with the operator .Φ ε + aI − ahk , which enjoys an improved regularity. The algorithm for solving the intermediate, regularized, VI is shown below.

.

Solving the Intermediate VI (6.71) in the PPM

Input. Fix a large number .a > 0 and a positive constant .α > 0. The k-th iterate from the PPM, .hk ∈ Λ, is known. Step 0. Initialization. Identify an initial feasible solution .h0 ∈ Λ, and set the iteration counter .n = 0. Step 1. Find the effective path delays. Solve the dynamic network loading subproblem with the vector of path departure rates given by .hn , and obtain ε n .Φp (t, h ), ∀t ∈ [t0 , tf ], p ∈ P. Step 2. Find the dual variable. For each .(i, j ) ∈ W, solve the following equation for .vij , using root-search algorithms.  

tf

.

p∈Pij

  hnp (t) − α Φpε (t, hn ) + ahn − ahk + vij dt = Qij



+

t0

Step 3. Update the path departure rates. For each .(i, j ) ∈ W and .p ∈ Pij , update the path departure rate as hn+1 p (t) =

.



  hnp (t) − α Φpε (t, hn ) + ahn − ahk + vij

+

∀t ∈ [t0 , tf ] (continued)

6.4 Convergence of Algorithms

225

Step 4. Check for convergence. If  n+1  h − hn L2   . ≤  hn  2 L

where . ∈ R++ is a prescribed termination threshold, then terminate the algorithm with output .hk+1 ≈ hn . Otherwise, set .n = n + 1 and repeat Step 1 through Step 4.

6.4 Convergence of Algorithms In this section, the convergence of the three algorithms will be analyzed and their conditions identified. The convergence of an algorithm is highly related to properties of the path delay operator, which is obtained through the dynamic network loading subproblem. In particular, most convergence proofs rest on certain types of continuity and generalized monotonicity of path delays. For example, Jang et al. (2005) develop a projection-based method to solve a route-choice DUE problem, which is a special case of the SRDC DUE problem; the convergence of this method requires continuity and strict monotonicity of the path delay operator. Similarly, a fixed-point method developed by Friesz et al. (2011) for SRDC DUEs relies on Lipschitz continuity and strong monotonicity. Strong monotonicity is known to be violated for general networks and DNL models (Mounce and Smith, 2007), and algorithms that rely on relaxed notions of monotonicity have been proposed in the literature. Lo and Szeto (2002a,b) develop an alternating direction method and a descent method for cell-based DUE problems with fixed and elastic demand, respectively; both methods require that the delay operator is co-coercive. According to Zhao and Hu (2007), a sufficient condition for co-coerciveness includes Lipschitz continuity and monotonicity (rather than strong monotonicity). Algorithms with even more relaxed convergence requirements, such as the day-to-day route swapping algorithm (Huang and Lam, 2002; Szeto and Lo, 2006; Tian et al., 2012) and the extragradient method (Long et al., 2013), are also proposed for solving DUE problems. In particular, convergence of the route swapping algorithm requires continuity and monotonicity (Mounce and Carey, 2011); and the extragradient method requires Lipschitz continuity pseudo monotonicity for convergence. This section provides the convergence theory for three proposed algorithms that extend the knowledge of mathematical tools available for ensuring convergence. Convergence of the fixed-point algorithm typically requires Lipschitz continuity and

226

6 Algorithms for Computing Dynamic User Equilibria

strong monotonicity of the path delay operator, except for elastic DUE, in which case convergence only requires weak monotonicity on part of the operator. The self-adaptive projection algorithm can be shown to converge given that the delay operator is continuous and pseudo monotone, which is a weaker notion than (strong) monotonicity. Finally, the proximal point method can be shown to converge for a class of quasi monotone delay operators, which further relaxes the monotonicity condition compared to the existing literature. In presenting our convergence results, we will adhere to the most generic (D)VI form of the problem without referring explicitly to the type of DUE models we are trying to solve. For example, the convergence conditions established for the proximal point method certainly apply to DUE with fixed demand, DUE with elastic demand, and DUE with bounded rationality, since all three DUE models can be expressed as generic VIs. Therefore, throughout this section, we consider a generic variational inequality problem:  .

F (x ∗ ), x − x ∗



≥ 0

∀x ∈ Ω

(6.72)

where F represents a general operator satisfying necessary regularity conditions and Ω is an arbitrary convex and closed set in a proper space X. We note that, however, depending on the type of DUE model the established convergence conditions have different implications on the effective delay operator, which relates to the dynamic network loading problem. Moreover, these convergence conditions may fail for general networks and traffic dynamics. Thus, the computational algorithms proposed here should be considered as heuristics when convergence cannot be rigorously assured by the underlying network model. Nevertheless, this section’s intent is to (i) document how far the available mathematics can take us in assuring convergence and (ii) illustrate what can be done computationally when proceeding heuristically by relaxing monotonicity assumptions needed to assure convergence.

.

6.4.1 Convergence Result for the Fixed-Point Algorithm 6.4.1.1

A Generic Proof Based on Strong Monotonicity

The convergence of the fixed-point algorithm requires Lipschitz continuity and strong monotonicity of the operator F . Definition 6.1 (Strong monotonicity) A mapping .F (·) is strongly monotone with constant .δ > 0 if .

F (x1 ) − F (x2 ), x1 − x2  ≥ δx1 − x2 2

∀x1 , x2 ∈ Ω

6.4 Convergence of Algorithms

227

Definition 6.2 (Lipschitz continuity) A mapping .F (·) is Lipschitz continuous with constant .L > 0 if F (x1 ) − F (x2 ) ≤ Lx1 − x2 

.

∀x1 , x2 ∈ Ω

Theorem 6.8 (Convergence of the fixed-point algorithm) Assume that the operator .F (·) is Lipschitz continuous with constant L and strongly monotone with constant .δ. Then the sequence .{x k } generated by the fixed-point algorithm converges to a solution of the VI (6.72). Proof The fixed-point iterative algorithm . x k+1 = PΩ [x k − αF (x k )] = G(x k )

.

converges if the operator .G(·) is non-expansive. We deduce from the nonexpansiveness of the minimum norm projection .PΩ [·] that G(x k+1 ) − G(x k )2

.

≤x k+1 − x k − αF (x k+1 ) + αF (x k )2 " ! =x k+1 − x k 2 − 2α F (x k+1 ) − F (x k ), x k+1 − x k + α 2 F (x k+1 ) − F (x k )2 ≤x k+1 − x k 2 − 2α · δx k+1 − x k 2 + α 2 · Lx k+1 − x k 2 =(1 − 2αδ + α 2 L)x k+1 − x k 2 Therefore, to ensure non-expansiveness, we must have 1 − 2αδ + α 2 L < 1 ⇒ 0 < α
0

.

is a positive scalar. Moreover, the following remark contains information vital to understanding the generality of our approach to convergence: Remark 6.2 Examination of (6.73) reveals that weakly monotone functions are quite general. Indeed, as shown in Hanet al. (2015), a sufficient condition for weak monotonicity is Lipschitz continuity, which is itself a very mild assumption. We will assume that the inverse demand function is strongly monotone decreasing. That is  2  

Θ 1 − Θ 2 , Q1 − Q2 E ≤ −1 · KΘ Q1 − Q2 

(6.74)

.

E

KΘ > 0

(6.75)

.

Such an assumption about inverse demand functions is behaviorally sound, assuring, in principle, that inverse demands “fall rapidly.” It follows from (6.73) and (6.74) that  2  

Ψ 1 − Ψ 2 , h1 − h2 L2 − Θ 1 − Θ 2 , Q1 − Q2 E ≥ −KΨ h1 − h2 

.

L2

 2   + KΘ Q1 − Q2 

E

(6.76) We additionally assume these forms of Lipschitz continuity: .

 2  2  k    Ψ − Ψ ∗  2 ≤ K1 hk − h∗  2 . L

 2  2  k    Θ − Θ ∗  ≤ K2 Qk − Q∗  E

E

K1 , K2 > 0

Q (h) =

tf





.

t0

.

(6.78) (6.79)

Moreover 

(6.77)

L

(i,j )∈W p∈Pij

hp (t)dt,

6.4 Convergence of Algorithms

229

where Q may be viewed as a functional of departure rate vector h, denoted by .Q (h). As a consequence, the Cauchy-Schwarz inequality leads us to .

 2  2  k    Q − Q∗  ≤ K0 hk − h∗  2 . E

(6.80)

L

K0 > 1,

(6.81)

which is recognized as another Lipschitz conitinuity condition. Let us now undertake to restate expression (6.76) in a fashion that will be helpful in proving convergence in the next section. In particular, where 0 < η < 1,

(6.82)

.

the right-hand side of (6.76) may be manipulated to yield  2  2  2       RH S = −KΨ hk − h∗  2 + ηKΘ Q1 − Q2  + (1 − η) KΘ Q1 − Q2 

.

L

E

.

E

(6.83)  2  2  2       ≤ −KΨ hk − h∗  2 + ηK0 KΘ hk − h∗  2 + (1 − η) KΘ Q1 − Q2  . L

L

E

(6.84) 2  2      = (ηK0 KΘ − KΨ ) hk − h∗  2 + (1 − η) KΘ Q1 − Q2  L

2 2       = A hk − h∗  2 + B Q1 − Q2  L

(6.85)

.

E

(6.86)

E

In (6.86), we have used these parameter definitions to simplify our notation: .

A ≡ (ηK0 KΘ − KΨ ) > 0.

(6.87)

B ≡ (1 − η) KΘ > 0,

(6.88)

where the strict inequality in (6.87) is an assumption that is now being introduced and will hold through the remainder of our presentation. The strict inequality in (6.88) follows from (6.82) and the strictly positive nature of .K0 , .KΘ , and .KΨ . The result is this statement of strong monotonicity:  2  

Ψ 1 − Ψ 2 , h1 − h2 L2 − Θ 1 − Θ 2 , Q1 − Q2 E ≥ A h1 − h2  2

.

L

2    + B Q1 − Q2 

E

(6.89)

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6 Algorithms for Computing Dynamic User Equilibria

Note that (6.89) implies

Ψ 1 − Ψ 2 , h1 − h2 L2 − Θ 1 − Θ 2 , Q1 − Q2 E > 0

.

(6.90)

provided h1 = h2

.

In other words, the operator  .

Ψ −1 · Θ



is strictly monotone increasing, thereby assuring an unique equilibrium solution  .

h∗ Q∗



for (4.33). We have shown in Theorem 6.2 that the E-DUE may be restated as a fixed-point problem. That formulation is $   # h h − αΨ (h) . = M (h, Q) , . = PΛ1 Q + αΘ (Q) Q

(6.91)

1 , which is defined where .PΛ1 [v] is the minimum norm projection of v onto the set .Λ in (6.11), while α>0

.

is an arbitrary positive scalar. We have already established that the fixed-point formulation (6.91) gives rise to this algorithm  .

hk+1 Qk+1



 $ hk − αΨ hk   , Qk + αΘ Qk

# = PΛ1

(6.92)

which is essentially the same as a projection algorithm for variational inequalities. The conventional manner of proving convergence of the projection-based fixedpoint algorithm involves showing .M (h, Q) defined in (6.91) is non-expansive for an appropriate choice of parameters associated with monotonicity, Lipschitz continuity, and the fixed-point formulation. Demonstrating non-expansiveness is promising since the minimum norm projection operator is known to be non-expansive in

6.4 Convergence of Algorithms

231

Hilbert spaces. We begin by observing the following:     .  2 Y k+1 =  hk+1 , Qk+1 − h∗ , Q∗ 

.

 k   ∗ 2   h − αΨ k h − αΨ ∗   − = k k ∗ ∗  Q + αΘ Q + αΘ  k     h − h ∗ − α Ψ k − Ψ ∗ 2     = Qk − Q∗ + α Θ k − Θ ∗  2  2      = hk − h∗  2 − 2α Ψ k − Ψ ∗ , hk − h∗ L2 + α 2 Ψ k − Ψ ∗  2 L

L

2  2      + Qk − Q∗  + 2α Θ k − Θ ∗ , Qk − Q∗ E + α 2 Θ k − Θ ∗  E E $ # = −2α Ψ k − Ψ ∗ , hk − h∗ L2 − Θ k − Θ ∗ , Qk − Q∗ E  2  2 2  2          + hk − h∗  2 + α 2 Ψ k − Ψ ∗  2 + Qk − Q∗  + α 2 Θ k − Θ ∗  L

L

E

E

(6.93) Using the Lipschitz continuity assumptions (6.77) and (6.78), together with the monotonicity assumptions (6.73) and (6.74), we obtain the following from (6.93): #  2 2 $      Y k+1 ≤ −2α A h1 − h2  2 + B Q1 − Q2 

.

L

E

 2  2     + hk − h∗  2 + α 2 K1 hk − h∗  2 L

L

2  2      + Qk − Q∗  + α 2 K2 Qk − Q∗  E

E

2 2        = 1 − 2αA + α 2 K1 hk − h∗  2 + 1 − 2αB + α 2 K2 Qk − Q∗  

L

E

(6.94) We assure that each iteration of the algorithm is a contraction by enforcing Y k+1 < Y k ,

.

which is guaranteed by requiring 1 − 2αA + α 2 K1 < 1

.

1 − 2αB + α 2 K2 < 1

(6.95)

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6 Algorithms for Computing Dynamic User Equilibria

which may be restated as .

α 2 K1 < 2αA.

(6.96)

α 2 K2 < 2αB

(6.97)

We now use definitions (6.87) and (6.88) to restate (6.96) and (6.97) as .

α 2 K1 < 2α (ηK0 KΘ − KΨ ) .

(6.98)

α 2 K2 < 2α (1 − η) KΘ

(6.99)

Thus, a list of all parametric inequalities we have invoked is the following KΨ < ηK0 KΘ .

.

αK1 + 2KΨ < ηK0 KΘ . 2 αK2 < (1 − η) KΘ . 2 K0 > 1. 1>η>0

(6.100) (6.101) (6.102) (6.103) (6.104)

Expression (6.100) is the previously introduced assumption (6.87). However, it should be noted (6.101) requires KΨ
0, with α ∈ (0, 1), the following holds: 

 x − x ∗ , αr(x, β) + βF (x − αr(x, β))    ≥ αr(x, β), r(x, β) − β F (x) − F (x − αr(x, β)) .

(6.109)

where .x ∗ is a solution of the original VI. Using (6.39) and (6.40), (6.109) is equivalent to  .

   x − x ∗ , d(x, β) ≥ r(x, β), g(x, β)

(6.110)

Proof According to (6.105) we have   β F (x − αr(x, β)), x − αr(x, β) − x ∗ ≥ 0

.

(6.111)

where .x ∗ ∈ Ω is a solution. Then setting .x = x − βF (x) and .y = x ∗ in (6.107) yields  .

 x − βF (x) − PΩ [x − βF (x)], x ∗ − PΩ [x − βF (x)] ≤ 0

(6.112)

That is,  .

 r(x, β) − βF (x), x ∗ + r(x, β) − x ≤ 0

(6.113)

Expressions (6.111) and (6.113) lead to the following calculations: 

 x − x ∗ , αr(x, β) + βF (x − αr(x, β))     ≥ αβ F (x − αr(x, β)), r(x, β) + α r(x, β), r(x, β) − βF (x)   + x − x ∗ , αβF (x)      = αr(x, β), r(x, β) − β F (x) − F (x − αr(x, β)) + αβ F (x), x − x ∗    ≥ αr(x, β), r(x, β) − β F (x) − F (x − αr(x, β)) .

where the last inequality uses (6.105).



Lemma 6.3 For any .x ∈ Ω and .0 < β1 < β2 , we have .

r(x, β1 ) ≤ r(x, β2 ) .

(6.114)

r(x, β2 ) r(x, β1 ) ≥ β1 β2

(6.115)

Proof The reader is referred to He (1995) for a proof.



6.4 Convergence of Algorithms

235

Lemma 6.4 Let .{x k }k≥1 ⊂ Ω be the sequence generated by the algorithm, then k .{x }k≥1 are bounded. Proof The proof is divided into several parts.

  Part 1. For each intermediate iterate .x k there holds .r(x k , 1) > 0. We claim that there exists .α¯ > 0 such that for every .0 < αk+1 ≤ α, ¯ there holds      βk F (x k )−F x k −αk+1 r(x k , βk )  ≤ Lr(x k , βk )

.

∀k ≥ 1

(6.116)

L where .L is the Lipschitz constant of .F . We then show To see this, we set .α¯ = L that (6.116) holds for every .0 < α < α. ¯ By contradiction, we have

       F (x k ) − F x k − αk+1 r(x k , βk )  > L r(x k , βk ) ≥ Lr(x k , 1) βk (6.117)

.

where the second inequality is a consequence of (6.115). With Lipschitz continuity, we deduce      F (x k ) − F x k − αk+1 r(x k , βk )  ≤ L αk+1 r(x k , βk )     ≤ L α¯ r(x k , βk ) ≤ Lr(x k , 1)

.

which is contradicting (6.117). The claim is established. Part 2. We deduce  .

 r(x k , βk ), g(x k , βk ) 2  = αk+1 r(x k , βk )   − αk+1 βk r(x k , βk ), F (x k ) − F (x k − αk+1 r(x k , βk )) 2  ≥ αk+1 r(x k , βk )      − αk+1 βk r(x k , βk ) · F (x k ) − F x k − αk+1 r(x k , βk )  2      ≥ αk+1 r(x k , βk ) − αk+1 r(x k , βk ) · Lr(x k , βk )  2 = αk+1 (1 − L)r(x k , βk ) (6.118)

where the first inequality uses the Cauchy-Schwarz inequality and the second inequality is due to (6.116). Recalling notation (6.39), the above is rewritten as  .

 2  x k − x ∗ , d(x k , βk ) ≥ αk+1 (1 − L)r(x k , βk )

(6.119)

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6 Algorithms for Computing Dynamic User Equilibria

Part 3.

Step 3 of the algorithm stipulates that   x k+1 = PΩ x k − γρ(x k , βk )d(x k , βk )

.

(6.120)

In addition, we have .PΩ [x ∗ ] = x ∗ since .x ∗ ∈ Ω. Thus, by the non-expansiveness of the projection operator, we have   k+1 2 2 x − x ∗  ≤ x k − γρ(x k , βk )d(x k , βk ) − x ∗   2   = x k − x ∗  − 2γρ(x k , βk ) x k − x ∗ , d(x k , βk ) . (6.121)   2 + γ 2 ρ(x k , βk )2 d(x k , βk )  2   ≤ x k − x ∗  − 2γρ(x k , βk ) r(x k , βk ), g(x k , βk )   + γ 2 ρ(x k , βk ) r(x k , βk ), g(x k , βk ) 2  ≤ x k − x ∗  − γ (2 − γ )ρ(x k , βk )  2 (6.122) · αk+1 (1 − L)r(x k , βk )

.

where the second inequality is due to (6.110) and the third inequality is due to (6.118). As a result, we have    k+1 2 2 2 x − x ∗  ≤ x k − x ∗  ≤ . . . ≤ x 0 − x ∗ 

.

Thus the sequence .{x k } is bounded.

(6.123)



We can now begin the proof of the main convergence theorem. Proof The proof is completed in two steps. Part 1. We show that .d(x k , βk ) is uniformly bounded for all .k ≥ 1. By definition,      d(x k , βk ) = αr(x k , βk ) + βk F x k − αr(x k , β)       ≤ αk+1 r(x k , βk ) + βk F x k − αr(x k , β) 

.

Notice thatsince path  kthe effective  delays are clearly uniformly bounded, so are k .F x − αr(x , β)  for all .k ≥ 1. Thus it remains to show that the norms   k .r(x , βk ) is uniformly bounded. Indeed, we deduce       PΩ [x k − βk F (x k )] − x ∗  ≤ PΩ [x k − βk F (x k )] − x ∗    = PΩ [x k − βk F (x k )] − PΩ [x ∗ ]

.

6.4 Convergence of Algorithms

237

  ≤ x k − βk F (x k ) − x ∗      ≤ x k − x ∗  + βk F (x k )

∀k ≥ 1

Thus, for all .k ≥ 1,          k   k k k  k k k . r(x , βk ) = x − PΩ [x − βk F (x )] ≤ x +PΩ [x −βk F (x )] ≤ M for some .M > 0. Part 2. From (6.122) we have that   k+1 2 2 x − x ∗  ≤ x k − x ∗  − γ (2 − γ )ρ(x k , βk )  2 · αk+1 (1 − L)r(x k , βk )  2 ≤ x k − x ∗  − γ (2 − γ )αk+1 (1 − L)   k  r(x , βk ), g(x k , βk )  k r(x , βk )2 · k d(x , βk ) .

  2 α 2 (1 − L)2  k r(x , βk )4 ≤ x k − x ∗  − γ (2 − γ ) k+1 2 M where the last inequality is due to (6.118). Thus, for any given .m ≥ 1, we use the above inequality recursively to get  m    0   m+1   αk+1 (1 − L) 2  ∗ 2 ∗ 2   r(x k , βk )4 . x −x ≤ x −x − γ (2 − γ ) M k=0

Thus γ (2 − γ )

.

 +∞      αk+1 (1 − L) 2  r(x k , βk )4 ≤ x 0 − x ∗ 2 < + ∞ M k=0

Given that .{αk+1 }k≥1 are uniformly bounded away from zero, i.e., by .α, ¯ we must have .

  lim r(x k , βk ) = 0

k→+∞

(6.124)

. We observe from Step 2 of the algorithm that .αk+1 ≥ αmin = min{α0 , μα}. ¯ Therefore, according to Step 1 of the algorithm, .βk = min{1, θ αk } ≥ . min{1, θ αmin } = βmin . Expressions (6.124) and (6.114) together imply .

  lim r(x k , βmin ) = 0

k→+∞

(6.125)

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6 Algorithms for Computing Dynamic User Equilibria

By Lemma 6.4, the sequence .{x k }k≥1 is bounded; thus one can extract a ˜ Clearly, convergent subsequence .{x kj } → x.   r(x, ˜ βmin ) =

.

  lim r(x kj , βmin ) = 0

j →+∞

implying that .x˜ is a solution. Replacing .x ∗ in (6.123) with .x˜ yields 2  2  2  k+1 x − x˜  ≤ x k − x˜  ≤ . . . ≤ x 0 − x˜ 

.

The fact that .x kj − x ˜ → 0 immediately yields .x k − x ˜ → 0 as .k → ∞.



6.4.3 Convergence Result for the Proximal Point Method We begin with the articulation of the dual formulation of the original VI. Definition 6.5 (Dual formulation of the VI) The dual form of the VI (6.72), also known as the Minty problem, is defined as follows. Find .x d ∈ Ω such that  .

 F (x), x − x d ≥ 0

∀x ∈ Ω

(6.126)

We let .Y d be the solution set of (6.126). Lemma 6.5 Assume that .Y d = ∅ and that .Ω is bounded with diameter .D < ∞. Then the sequence .{x k } generated by the proximal point method satisfies the following:  .

 aD 2 F (x μ(k)+1 ) , x − x μ(k)+1 ≥ − √ k+1

∀x ∈ Ω

(6.127)

. where .μ(k) = argmin x i − x i+1 2 ∈ {0, . . . , k}. 0≤i≤k

Proof Let .x d ∈ Y d . We have that for any .k ≥ 0,   x k −x d 2 = x k+1 −x k 2 +x d −x k+1 2 +2 x k+1 −x k , x d −x k+1

.

(6.128)

Taking .x = x d in (6.63) and combining this with (6.126) yields     0 ≥ F (x k+1 ), x d − x k+1 ≥ − a x k+1 − x k , x d − x k+1

.

(6.129)

6.4 Convergence of Algorithms

239

Equations (6.128) and (6.129) together yield that for all .k ≥ 0 x k+1 − x k 2 ≤ x k − x d 2 − x k+1 − x d 2

(6.130)

.

Summing up (6.130) for different values of k yields (k + 1) min x i+1 − x i 2 ≤

k 

.

0≤i≤k

x i+1 − x i 2 ≤ x 0 − x d 2 − x k+1 − x d 2

i=0

(6.131) . For each .k ≥ 0, introduce the notation .μ(k) = argmin x i − x i+1 2 . Then (6.131) 0≤i≤k

becomes (k + 1)x μ(k)+1 − x μ(k) 2 ≤ x 0 − x d 2 ≤ D 2

.

and thus x μ(k)+1 − x μ(k) 2 ≤

.

1 D2 k+1

(6.132)

 By invoking (6.63), we have where D denotes the diameter of the bounded set .Λ.  .

 F (x μ(k)+1 + a(x μ(k)+1 − x μ(k) ) , x − x μ(k)+1 ≥ 0

 ∀x ∈ Λ,

which immediately leads to  .

   F (x μ(k)+1 ) , x − x μ(k)+1 ≥ − a x μ(k)+1 − x μ(k) , x − x μ(k)+1 ≥ − ax μ(k)+1 − x μ(k)  · x − x μ(k)+1  ≥ − ax μ(k)+1 − x μ(k)  · D % 1  D2 · D ≥ −a ∀x ∈ Λ k+1

(6.133)

 As an immediate consequence of Lemma 6.5, we have the following convergence result. Theorem 6.11 (Convergence of the proximal point method) Assume that .Y d =  ∅ and that .Ω is bounded with diameter .D < ∞. Then for any tolerance .δ > 0, 2 4 . there exists .R =  a δD2  − 1, such that  .

 F (x μ(R)+1 ) , x − x μ(R)+1 ≥ − δ

∀x ∈ Ω,

(6.134)

240

6 Algorithms for Computing Dynamic User Equilibria

where .{x k }, k ≥ 0, is the sequence generated by the proximal point method and .z denotes the smallest integer that is larger than or equal to z. Moreover, when the δ PPM algorithm terminates, i.e., when .x k+1 − x k  ≤ aD for the first time, then  .

 F (x k+1 ) , x − x k+1 ≥ − δ 2

∀x ∈ Ω

(6.135)

4

Proof Setting .k = R =  a δD2  − 1 in (6.127) yields  .

 F (x μ(R)+1 ) , x − x μ(R)+1 ≥ − δ

∀x ∈ Ω

According to the definition of .μ(·), the termination criterion of the PPM implies that .k = μ(k). In addition, by (6.133), we have  .

 F (x μ(k)+1 ) , x − x μ(k)+1 ≥ − aDx μ(k)+1 − x μ(k) 

Equation (6.135) follows immediately from the fact that .k = μ(k) and .x k+1 − δ x k  ≤ aD .  Remark 6.3 Unlike the convergence results established for the fixed-point and selfadaptive projection algorithms, which focus on the asymptotic behavior of the sequence .{x k } as .k → ∞, the convergence result for the proximal point method is concerned with finding a solution of the approximate VI (i.e., with .−δ on the right-hand side) within a finite number of iterations. Such a convergence result is quite practical for numerical computations as it estimates the number of iterations needed to achieve a given level of accuracy in approximating the solution of the original VI. Theorem 6.11 only requires that the dual VI (6.126) has a solution—a property subsequently referred to as dual solvability. Compared to the convergence conditions for the self-adaptive projection method (Sect. 6.4.2), dual solvability is weaker than assumption (6.105), as the latter requires that the solution of the original VI must be a solution of the dual VI. In addition, Theorem 6.11 does not rely on the continuity of the principal operator. We thus conclude that the convergence conditions for the PPM are indeed weaker than those of the previous two methods. In the remainder of this subsection, we will investigate in detail dual solvability and provide sufficient conditions for it. One should note that if the original VI has a solution, then a sufficient condition for dual solvability is pseudo monotonicity; this is apparent from Definition 6.4. In the following presentation, we will articulate a weaker sufficient condition for dual solvability, based on the notion of semistrictly quasi monotonicity. Definition 6.6 The operator .F is quasi monotone if, for arbitrary .x 1 , x 2 ∈ Ω, ! .

" ! " F (x 2 ) , x 1 − x 2 > 0 ⇒ F (x 1 ) , x 1 − x 2 ≥ 0

References and Suggested Reading

241

The operator .F is semistrictly quasimonotone if it is quasimonotone and, for every x 1 , x 2 ∈ Ω, ! " ! " 2 1 2 . F (x ) , x − x > 0 ⇒ F (x 3 ) , x 1 − x 2 > 0

.

  for some .x 3 ∈ x : x = x 1 + λ(x 2 − x 1 ), λ ∈ (0, 1/2) . The reader is referred to Konnov (1998) for a detailed discussion of quasimonotonicity. In particular, Lemma 3.1 of Konnov (1998) states that pseudomonotonicity implies semistrictly quasimonotonicity; thus the latter is a weaker assumption. We also need to define .w ∗ -hemicontinuity as below. Definition 6.7 .F is .w ∗ -hemicontinuous if the function ! " . .c(λ) = F (xλ ) , xλ − x 2 , where xλ = λx 1 + (1 − λ)x 3 is upper semicontinuous at .λ = 0+ for all .x 1 , x 2 , x 3 ∈ Ω and .λ ∈ [0, 1]. Finally, the sufficient condition for dual solvability is summarized below. Theorem 6.12 (Sufficient condition for dual solvability) If .F is continuous on .Ω and is semistrictly quasimonotone, then the dual problem (6.126) has a solution. Proof It is easy to verify by definition that if .F is continuous in the strong topology, then it is .w ∗ -hemicontinuous. Thus the conclusion follows from Theorem 4.1 of Konnov (1998).  We offer the following concluding remarks. The three computational methods proposed in this chapter rely on generalized monotonicity in order to converge. As we previously mentioned, delay operators may not satisfy these generalized notions of monotonicity, with only a few exceptions (Mounce, 2006; Perakis and Roels, 2006). Thus the proper perspective on convergence of numerical algorithms for calculating DUEs on general networks is to say that almost all algorithms are presently heuristic. Exact algorithms will be created only when a fundamentally new operator class is invented, which allows non-monotonicity while also providing behavioral insights that allow convergence to be established.

References and Suggested Reading Allevi, E., Gnudi, A., & Konnov, I. V. (2006). The proximal point method for nonmonotone variational inequalities. Mathematical Methods of Operations Research, 63, 553–565. El Farouq, N. (2001). Pseudomonotone variational inequalities: convergence of proximal methods. Journal of optimization theory and applications, 109(2), 311–326. Friesz, T. L. (2010). Dynamic optimization and differential games. New York: Springer.

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Friesz, T. L., Bernstein, D., Smith, T., Tobin, R., & Wie, B. (1993). A variational inequality formulation of the dynamic network user equilibrium problem. Operations Research, 41(1), 80–91. Friesz, T. L., Bagherzadeh, A. & Han, K. (2021). Convergence of fixed-point algorithms for elastic demand dynamic user equilibrium. Transportation Research Part B, 150, 336–352. Friesz, T. L., Han, K., Neto, P., Meimand, A., & Yao, T. (2013). Dynamic user equilibrium based on a hydrodynamic model. Transportation Research Part B, 47(1), 102–126. Friesz, T. L., Kim, T., Kwon, C., & Rigdon, M. A. (2011). Approximate network loading and dual-time-scale dynamic user equilibrium. Transportation Research Part B, 45(1), 176–207. Han, D., & Lo, H. K., (2002). Two new self-adaptive projection methods for variational inequality problems. Computers and Mathematics with Applications, 43, 1529–1537. Han, K., Friesz, T. L., Szeto, W. Y., & Liu, H. (2015). Elastic demand dynamic network user equilibrium: Formulation, existence and computation. Transportation Research Part B: Methodological, 81, 183–209. He, B. S. (1995). Some predict-correct projection methods for monotone variational inequalities. Reports of the Faculty of Technical Mathematics and Informatics, pp. 95–68, Delft. Huang, H. J., & Lam, W. H. K. (2002). Modeling and solving the dynamic user equilibrium route and departure time choice problem in network with queues. Transportation Research Part B, 36(3), 253–273. Jang, W., Ran, B., & Choi, K. (2005). A discrete time dynamic flow model and a formulation and solution method for dynamic route choice. Transportation Research Part B, 39(7), 593–620. Konnov, I. V., 1998. On quasimonotone variational inequalities. Journal of Optimization Theory and Applications, 99(1), 165–181. Konnov, I. V., (2003). Application of the proximal point method to non monotone equilibrium problems. Journal of Optimization Theory and Applications, 119, 317–333. Lo, H., & Szeto, W. Y. (2002a). A cell-based variational inequality formulation of the dynamic user optimal assignment problem. Transportation Research Part B, 36(5), 421–443. Lo, H., & Szeto, W. Y. (2002b). A cell-based dynamic traffic assignment model: formulation and properties. Mathematical and Computer Modelling, 35(7–8), 849–865. Long, J. C., Huang H. J., Gao, Z. Y., & Szeto, W. Y. (2013). An intersection-movement-based dynamic user optimal route choice problem. Operations Research, 61(5), 1134–1147. Mounce, R. (2006). Convergence in a continuous dynamic queuing model for traffic networks. Transportation Research Part B, 40(9), 779–791. Mounce, R., & Carey, M. (2011). Route swapping in dynamic traffic networks. Transportation Research Part B, 45(1), 102–111. Mounce, R., & Smith, M. (2007). Uniqueness of equilibrium in steady state and dynamic traffic networks. In R. E.Allsop, M. G. H. Bell & B. G. Heydecker (Eds.), Transportation and traffic theory (pp. 281–299). Elsevier. Perakis, G., & Roels, G. (2006). An analytical model for traffic delays and the dynamic user equilibrium problem. Operations Research, 54(6), 1151–1171. Szeto, W. Y., & Lo, H. K. (2006). Dynamic traffic assignment: Properties and extensions. Transportmetrica, 2(1), 31–52. Tian, L. J., Huang, H. J., & Gao, Z. Y. (2012). A cumulative perceived value-based dynamic user equilibrium model considering the travelers’ risk evaluation on arrival time. Networks and Spatial Economics, 12(4), 589–608. Zhao, Y., & Hu, J. (2007). Global bounds for the distance to solutions of co-coercive variational inequalities. Operations Research Letters, 35(3), 409–415.

Chapter 7

Dynamic Network Loading: Non-physical Queue Models

An essential component of many DTA models is a procedure known as dynamic network loading (DNL). The DNL subproblem aims at describing and predicting the spatial-temporal evolution of traffic flows on a network that is consistent with established route and departure time choices of travelers, by introducing appropriate dynamics to flow propagation, flow conservation, and travel delays on a network level. Any DNL must be consistent with the established path flows and link delay model, and DNL is usually performed under the first-in-first-out (FIFO) rule. Dynamic network loading relates closely to the path effective delay operator, which plays a pivotal role in DTA, especially DUE, problems. The properties of the effective delay operator are critical to proving existence and uniqueness of a solution to the infinite-dimensional variational inequality used to express DUE; they also affect convergence of any type of numerical algorithm used to compute DUE. The delay operator is usually not available in closed form and has to be numerically evaluated via the DNL procedure. The next two chapters present a detailed description of several DNL models and their underlying components such as link dynamics, junction model, and link delay functions. In presenting these models, we take a common perspective of distinguishing between non-physical queue models (Chap. 7) and physical queue models (Chap. 8). In traffic flow modeling, non-physical queue models usually mean that any link has infinite capacity in holding vehicles, so that spillback does not occur. Spillback is a traffic phenomenon frequently observed and reported in heavily congested networks where a queue (or a congested region) on a specific link exceeds the length of that link and spills over into upstream links, blocking vehicles on other links. Spillback is a typical example of the interdependence of congestion on different links; it also causes the fast propagation of congestion through a network and, in some extreme cases, traffic gridlock.

© Springer Nature Switzerland AG 2022 T. L. Friesz, K. Han, Dynamic Network User Equilibrium, Complex Networks and Dynamic Systems 5, https://doi.org/10.1007/978-3-031-25564-9_7

243

244

7 Dynamic Network Loading: Non-physical Queue Models

Non-physical queue models assume, either directly or indirectly, that vehicle queues have negligible sizes and hence will not affect traffic conditions outside of the current link.

7.1 The Link Delay Model One way to model path delay in dynamic networks is proposed by Friesz et al. (1993), who employ a volume-dependent link traversal time function. Such a perspective on path delay has been referred to as the link delay model (LDM). The link delay model presented in Friesz et al. (1993) belongs to the more general class of link models known as explicit exit time (or delay) function models. In these models, the link traversal time is expressed as an explicit function of certain state variable like link volume and entering/exiting flow. Such models are not always consistent with the first-in-first-out (FIFO) principle, depending on the explicit travel time function employed. For example, if the delay time function is assumed to depend on the link volume alone, then affine delay functions, i.e., functions of the form D(t) = αx(t) + β

.

α, β ∈ R++

are the only delay functions currently known that guarantee FIFO under any inflow profile. The link delay model with an affine delay function, as first articulated in Friesz et al. (1993), has been studied extensively in the context of dynamic traffic assignment (Astarita, 1996; Carey & McCartney, 2002; Ran & Boyce, 1996; Wu et al., 1998; Xu et al., 1999; Zhu & Marcotte, 2000).

7.1.1 Link Dynamics The link delay function originally proposed by Friesz et al. (1993) takes the following form D[x(t)] = αx(t) + β

.

∀α, β ∈ R++

(7.1)

where .x(t) denotes the traffic volume on the link at the entry time t. The quantity D[x(t)] represents the travel time on the link when the time of entry is t. The easiest way to interpret such delay functions is to treat each arc as if it has a fixed travel time .β and a congestion-related queuing time that is determined by the service rate .1/α. The first result to establish in this section is the monotonicity of the arc exit time function .τ (·) defined as .

7.1 The Link Delay Model

245

τ (t) = t + D[x(t)] = t + αx(t) + β

.

where .x(t), the link occupancy, satisfies .

d x(t) = u(t) − v(t) dt

(7.2)

where .u(·) and .v(·) are link entry flow and exit flow, respectively. Equation (7.2) is a simple consequence of vehicle conservation. Once the monotonicity of .τ (·) is shown, the FIFO rule will be automatically true. To simplify the notation, we will, in this section, let .D(t) denote the delay when entering the arc of interest at t. The following lemma is needed to demonstrate the main result. Lemma 7.1 For any differentiable, invertible function .f : R → R with derivative f :

.

      f −1 f (t) ≡ t ⇒ f −1 (z) ≡ 1/f  f −1 (z)

.

(7.3)

Proof Simply observe that       f −1 f (t) ≡ t ⇒ f −1 f (t) · f  (t) ≡ 1     ⇒ f −1 (z) · f  f −1 (z) ≡ 1, z = f (t)     ⇒ f −1 (z) ≡ 1/f  f −1 (z)

.

 We are ready to demonstrate the following theorem. Theorem 7.1 For any affine arc delay function of the form (7.1), the resulting arc exit time function .τ (·) is strictly increasing, and hence the inverse function .τ −1 (·) exists. Proof We proceed by partitioning time into appropriate intervals. In particular, assume, without loss of generality, that .x(0) = 0 and the first vehicle enters the arc of interest at time 0. Furthermore, let .t1 denote the time that the first vehicle exits the arc of interest. Then, by definition t1 = D(0) = β

.

(7.4)

and, for all .t ∈ [0, t1 ], 

t

x(t) =

u(s) ds

.

0

(7.5)

246

7 Dynamic Network Loading: Non-physical Queue Models

Hence, for each .t ∈ [0, t1 ], the exit time .τ1 (t) = τ (t) is given by  τ1 (t) = t + D(t) = t + α

.

t

u(s) ds + β

(7.6)

0

so that, in particular, .t1 = τ1 (0). Therefore, .τ1 (·) is differentiable with τ1 (t) = 1 + αu(t) > 0

.

(7.7)

and .τ (·) is increasing on .[0, t1 ]. So, if we now let t2 = τ1 (t1 )

.

(7.8)

then on the interval .[t1 , t2 ] = [τ1 (0), τ1 (t1 )], there must exit a well-defined inverse function .τ1−1 : [t1 , t2 ] → [0, t1 ]. Next, observe that at any time .t ∈ [t1 , t2 ] the commuters currently on the arc at time t are precisely those who entered before time t but have not yet exited (i.e., who entered during the interval .[τ1−1 (t), t]). Hence, if the exit time for commuters entering during the interval .[t1 , t2 ] is denoted by .τ2 , then for all .t ∈ [t1 , t2 ]  τ2 (t) = t + α

.

t τ1−1 (t)

u(s) ds + β

(7.9)

Note also from (7.6), (7.8), and (7.9) that τ2 (t1 ) = τ1 (t1 ) = t2

.

(7.10)

So, by differentiating the integral in (7.9) and applying both (7.3) and (7.7), we obtain    τ2 (t) = 1 + αu(t) − αu τ1−1 (t) τ1−1 (t)   αu τ1−1 (t) = 1 + αu(t) −   −1  τ1 τ1 (t)   αu τ1−1 (t) = 1 + αu(t) −   1 + αu τ1−1 (t)

.

= αu(t) +

1  −1  1 + αu τ1 (t)

> αu(t) ≥ 0

(7.11)

7.1 The Link Delay Model

247

Thus, we may conclude that .τ2 is also strictly increasing, which again implies that the inverse function τ2−1 : [τ2 (t1 ), τ2 (t2 )] → [t1 , t2 ]

.

is well-defined. We now proceed by induction as follows. Choose any .n ≥ 2, and suppose that as an extension of (7.9) and (7.10) for each .k = 2, . . . , n, there exist invertible functions .τk : [tk−1 , tk ] → [τk (tk−1 ), τk (tk )] satisfying the following three conditions for all .k = 2, . . . , n  τk (t) = t + α

.

t −1 τk−1 (t)

u(s) ds + β

∀t ∈ [tk−1 , tk ]

τk−1 (tk ) = τk (tk )

(7.13)

.

τk (t) > αu(t)

.

(7.12)

∀t ∈ [tk−1 , tk ]

(7.14)

Under these hypothesis, we wish to show that if .tn+1 = τn (tn ), then the function τn+1 : [tn , tn+1 ] → [τn+1 (tn ), τn+1 (tn+1 )]

.

defined by  τn+1 (t) = t + α

.

t

τn−1 (t)

u(s) ds + β

∀t ∈ [tn , tn+1 ]

(7.15)

also satisfies the following conditions paralleling (7.13) and (7.14) τn+1 (tn ) = τn (tn ).

(7.16)

.

 τn+1 (t)

> αu(t)

∀t ∈ [tn , tn+1 ]

(7.17)

To do so, observe (from the same argument as in (7.11)) that  .τn+1 (t)

  αu τn−1 (t) = 1 + αu(t) −  −1  τn τn (t)

(7.18)

Moreover, by (7.13) and the hypothesized invertibility of the function .τk , k = 2, . . . , n, it follows that τn (tn−1 ) = tn ⇒ tn−1 = τn−1 (tn )

.

248

7 Dynamic Network Loading: Non-physical Queue Models

and −1 tn = τn−1 (tn−1 ) ⇒ τn−1 (tn ) = tn−1

.

This together with (7.15) yields  τn+1 (tn ) = tn + α

.

 = tn + α  = tn + α

tn τn−1 (tn ) tn

u(s) ds + β

u(s) ds + β

tn−1 tn −1 τn−1 (tn )

u(s) ds + β

= τn (tn )

(7.19)

so that (7.16) is seen to hold. −1 Finally, since the invertibility of .τn implies  that .τn (t) ∈ [tn−1 , tn ] for all .t ∈  −1 [tn , tn+1 ], we see from (7.14) that .τn τn (t) is well-defined for all .t ∈ [tn , tn+1 ]. Hence, we may conclude from (7.18) that 

 .τn+1 (t)

  αu τn−1 (t) = αu(t) + 1 −  −1  > αu(t) τn τn (t)

(7.20)

so (7.17) is also seen to hold. Thus, the exit function .τ : R+ → R+ must be continuous and  increasing on each interval .[tn , tn+1 ], n ≥ 0. But, since the combined interval . n≥0 [tn , tn+1 ] is connected and tn+1 − tn = τn+1 (tn ) − tn ≥ β > 0

.

we may conclude that .R+ = increasing on .R+ .



n≥0 [tn , tn+1 ]. Thus, .τ

∀n ≥ 0

is everywhere continuous and 

7.1.2 Network Extension Friesz et al. (1993) introduced the notion of exit time functions together with a variational inequality to describe dynamic user equilibrium; that model is consistent with FIFO for appropriate arc delay functions, even though explicit flow propagation p constraints are not employed. In particular, they introduce a function τai (t) that

7.1 The Link Delay Model

249

expresses the time of exit from arc ai of every path p = {a1 , a2 , . . . , ai−1 , ai , ai+1 , . . . , am(p) } ∈ P

.

where P is the set of all network paths. The exit time functions obey the recursive relationships p

τa1 (t) = t + Da1 [xa1 (t)] ∀p ∈ P.    p τai (t) = τai−1 (t) + Dai xai τai−1 (t)

.

(7.21) ∀p ∈ P, i ∈ {2, . . . , m(p)} (7.22)

where Dai [xai (t)] is the time to traverse arc ai ; it is a function of the number of vehicles xai in front of the entering vehicle at the time of entry. Friesz et al. (2001) employed path delays computed from (7.21) and (7.22) with dynamics p

.

dxa1 (t) p = hp (t) − va1 (t) dt

∀p ∈ P.

(7.23)

p

dxai (t) p p = vai−1 (t) − vai (t) dt

∀p ∈ P, i ∈ {2, . . . , m(p)}

(7.24) p

where hp (t) is the departure rate at the beginning of path p ∈ P, xai (t) is the p volume of traffic on arc ai of path p for i ∈ {1, . . . , m(p)}, and vai (t) denotes the flow exiting that same arc. The arc volume xai (t) appearing in (7.21) and (7.22) is related to the path-specific p arc volume xai (t) via xa (t) =



.

p

δap xa (t)

∀a ∈ A

(7.25)

p∈P

where A is the set of arcs. Expression (7.25) makes use of the arc-path incidence matrix Δ =

.

  δap

where  δap =

.

1 if a ∈ p 0 otherwise

A fundamental flow propagation constraint, which relates arc delay to arc entering and exiting flow, is suggested by Bernstein et al. (1993), Ran et al. (1996), and Ran and Boyce (1996) and takes the form  p p Ua (t) = Va t + Da [x(t)]

.

∀a ∈ A, p ∈ P.

(7.26)

250

7 Dynamic Network Loading: Non-physical Queue Models

  Ua (t) = Va t + Da [x(t)] p

∀a ∈ A

(7.27)

p

where Ua (·) and Va (·) are the cumulative numbers of vehicles associated with path p that are entering and leaving link a, respectively. The meanings of notations Ua (·) and Va (·) as aggregates are made clear subsequently. The meaning of these constraints is fairly intuitive: vehicles entering an arc at a given moment in time must exit at a later time consistent with the arc traversal time. Notice that (7.26) and (7.27) are consistent with path-FIFO and link-FIFO, respectively. p p If we denote ua (·), va (·) to be the inflow and exit flow of arc a associated with path p, respectively, and ua (t) =



.

p

δap ua (t),

va (t) =

p∈P



p

δap va (t)

p∈P

then the following is true by definition 

p

t

Ua (t) =

.

p

ua (s) ds,

p



 Ua (t) =

0



t

ua (s) ds, 0

p

p

va (s) ds

0

.

t

Va (t) = Va (t) =

t

va (s) ds 0

p

Since Ua , Va , Ua , Va are absolutely continuous and, hence, differentiable almost everywhere, we obtain by differentiating (7.26) and (7.27) that

 p p ua (t) = va t + Da [x(t)] · 1 + Da [xa (t)]x˙a .

  ua (t) = va t + Da [x(t)] · 1 + Da [xa (t)]x˙a

.

(7.28) (7.29)

which are the flow propagation constraints employed by Friesz et al. (2001) in their formulation.

7.1.3 Formulation of the Dynamic Network Loading Problem Knowledge of the flow propagation constraints allow one to articulate the following differential algebraic equation (DAE) system describing dynamic network loading: p

.

dxai (t) p p = vai−1 (t) − vai (t) dt p

p,0

xai (0) = xai ∈ R+

∀p ∈ P, i ∈ {1, . . . , m(p)}.

∀p ∈ P, i ∈ {1, m(p)}.

(7.30) (7.31)

7.1 The Link Delay Model

251

  hp (t) = va1 t + Da1 [xa1 (t)] 1 + Da 1 [xa1 (t)] x˙a1 .

 p p vai−1 (t) = vai t + Dai [xai (t)] 1 + Da i [xai (t)] x˙ai (t) ∀p ∈ P, i ∈ {2, . . . , m(p)}

(7.32)

(7.33)

p

where .vai (t) is the flow along path p that exits arc .ai at time t. By convention p

va0 (t) ≡ hp (t)

.

∀p ∈ P

is the departure rate (path flow) from the origin of path .p ∈ P. An immediate consequence of the recursive relationships (7.21) and (7.22) is that the total traversal time for path p can be articulated in terms of the final exit time function and the departure time m(p)

Dp (t, h) =

.

  p p p τai (t) − τai−1 (t) = τam(p) (t) − t

∀p ∈ P

(7.34)

i=1

where .Dp (t, h) is the path traversal time for p, when the departure time is t, and p τai (t) is the time of exit from arc .ai , .i ∈ {1, . . . , m(p)} for path p given departure from the origin at t. If the vector of path departure rates .h(·) is given, upon solving the system of DAEs (7.30)–(7.33), one can find the arc exit flows and volumes. Let us denote the traffic volumes from solution of that system by

.

 p  x = xai : i ∈ {1, . . . , m(p)}, p ∈ P

.

and define the arc volume xa (t) =



.

p

δap xa (t)

∀a ∈ A

p∈P

In view of the affine delay function (7.1), we write the arc exit function of the first arc as p

τa1 (t) = t + Da1 [xa1 (t)] = t + αa1 xa1 (t) + βa1

.

where t is the departure time. Once the arc exit time function of the first arc has been computed, the arc exit time function for the next arc in the path may be computed as p

p

p

τa2 (t) = τa1 (t) + Da2 [xa2 (τa1 (t))]

.

252

7 Dynamic Network Loading: Non-physical Queue Models

and so forth until the arc exit times of all arcs have been computed. This procedure is carried out for each path .p ∈ P. Then the path delay can be computed according to (7.34). On top of the path delay operator, we introduce the effective path delay operator which generalizes the notion of travel cost to include early or late arrival penalties. We consider the effective path delays of the following form.  Ψp (t, h) = Dp (t, h) + f t + Dp (t, h) − TA

.

∀t ∈ [t0 , tf ], p ∈ P

(7.35)

where .TA is the target arrival time.

7.1.4 Continuity of the Effective Path Delay Operator The main result of this section, to be established in Theorem 7.2, is the strong continuity of the effective path delay operator .Ψ for a general road network, whose link dynamics are described by the link delay model. Theorem 7.2 Consider a general network .(A, V), where arc dynamics are governed by the link delay model; assume the link delay function for each .a ∈ A is affine. That is Da [xa (t)] = αa Xa (t) + βa

.

where .αa ∈ R1+ and .βa ∈ R1++ . Then the effective delay operator from .Λ into  2 |P | . L ([t0 , tf ] : h ∈ Λ −→ Ψ (·, h) is a continuous map. Remark 7.1 In Zhu and Marcotte (2000), the authors show that the effective delay operator is weakly continuous when the LDM is employed, under the restrictive assumption that the path flows are a priori bounded from above. That assumption is dropped in our result; we also assert strong, not weak continuity. The above theorem tells us there are no jump or other kinds of discontinuities of the path delay operator. Such an analytical result is crucial for the study of existence of dynamic user equilibrium and for analyzing DUE algorithms, when the dynamic network loading is based on the link delay model. Let us now present the proof of Theorem 7.2. Proof We begin by showing that given a converging sequence .h(n) in the space

|P | 2 .Λ ⊂ L+ ([t0 , tf ]) such that   (n) h − h

.

L2

−→ 0

n −→ ∞,

(7.36)

    the corresponding delay function .Dp ·, h(n) converges uniformly to .Dp ·, h for all .p ∈ P. This will be proved in several steps.

7.1 The Link Delay Model

253

Part 1. First, let us consider just one single arc, and hence omit the subscript a for brevity. Assume a sequence of entering flows .{u(n) }n≥1 converging to u in the 2 .L ([t0 , tf ]) space; that is   (n) u − u

.

L2

. =



tf



2 1/2 u(n) (t) − u(t) dt −→ 0

n −→ ∞

t0

(7.37)

Define the cumulative entering vehicle counts  t . U (n) (t) = u(n) (s) ds t  t 0 . . U (t) = u(s) ds

n ≥ 1

t ∈ [t0 , tf ]

t0

Then we assert the uniform convergence .U (n) −→ U on .[t0 , tf ]. To see this, fix any .ε > 0, in view of (7.37); choose .N > 0 such that for all .n > N   (n) u − u

.

L2

< ε

According to the embedding of .L1 ([t0 , tf ]) into .L2 ([t0 , tf ]), we deduce for any .t ∈ [t0 , tf ] that  t   t     (n)  U (t) − U (t) =  u(n) (s) ds − u(s) ds  

.

t0

t0

    ≤ u(n) − u1 ≤ (t0 − tf )1/2 u(n) − uL2 < (t0 − tf )1/2 ε The preceding shows the uniform convergence .U (n) −→ U on .[t0 , tf ]. Part 2. We adapt the recursive technique devised in Friesz et al. (1993). Let (n) (·), .n ≥ 1, and .X(·) denote the arc volumes corresponding to .U (n) (·), n ≥ .X 1, and .U (·), respectively. Assume, without loss of generality, that X(n) (t0 ) = 0,

.

X(t0 ) = 0

and that, for the flow profile .U (·), the first vehicle enters the arc of interest at time t0 . In addition, let .t1 denote the time that first vehicle exits the arc of interest. By definition

.

t1 = D(0) = β

.

(7.38)

254

7 Dynamic Network Loading: Non-physical Queue Models

For all .t ∈ [t0 , t1 ], since no vehicle can exit the arc before time .t1 , we have X(n) (t) = U (n) (t)

X(t) = U (t)

.

t ∈ [t0 , t1 ]

(7.39)

For each flow profile .U (n) , n ≥ 1, denote the exit time function restricted to (n) .[t0 , t1 ] by .τ 1 (·); under flow profile U , denote the exit time function restricted to .[t0 , t1 ] by .τ1 (·). Then   (n) τ1 (t) = t + D X(n) (t) = t + a U (n) (t) + β t ∈ [t0 , t1 ].   t ∈ [t0 , t1 ] τ1 (t) = t + D X(t) = t + a U (t) + β

.

(n)

We conclude that .τ1

(7.40) (7.41)

−→ τ1 uniformly on .[t0 , t1 ]. Now let . (n) t˜2 = inf τ1 (t1 ) ≤ τ1 (t1 )

.

n

 (n) −1 . Fix .δ small enough and call .t2 = t˜2 − δ. See Fig. 7.1. By Theorem 7.1, . τ1 , −1 .n ≥ 1, and .τ are well-defined, continuous, and strictly increasing. We claim  (n) −11 }n≥1 uniformly converges to .τ1−1 on .[t1 , t2 ]. To see this, we need that .{ τ1 to extend the arrival time functions .τ1 and .τ1(n) to the interval .(−∞, t0 ). Because

τ1(n) τ1

~ t ~ 2 t2 = t2 −δ

t1

t0

t1

Fig. 7.1 Proof of strong continuity of .Ψ , when the link delay model is employed: definitions of .t˜2 and .t2

7.1 The Link Delay Model

255

no vehicle is present during .(−∞, t0 ), it is natural to assign τ1 (t) = t + β,

.

τ (n) (t) = t + β

This means, if an infinitesimal flow particle enters the arc at .t ∈ (−∞, t0 ), its travel delay will always be .β. Fix any .ε < t1 − τ1−1 (t2 ), and consider the following quantities . Δ− ε =

.

inf

t∈[t0 , τ1−1 (t2 )]

  τ1 (t)−τ1 (t−ε)

. Δ+ ε =

inf

t∈[t0 , τ1−1 (t2 )]

  τ1 (t+ε)−τ1 (t) (7.42)

Since the infimum of a continuous function on a compact interval must be + obtained at some point .t ∈ [t0 , τ1−1 (t2 )], we conclude .Δ− ε , Δε > 0 by the strict monotonicity of .τ1 established in Theorem 7.1. (n) According to the uniform convergence .τ1 −→ τ1 on .(−∞, t1 ], there exists some .N > 0 such that as soon as .n ≥ N, we have τ1 (t) ≤ τ1 (t) + Δ− ε /2

.

(n)

τ1 (t) ≥ τ1 (t) − Δ+ ε /2 (n)

∀t ∈ [−∞, t1 ] (7.43)

For any .s ∈ [t1 , t2 ], we have .τ1−1 (s) ∈ [t0 , τ1−1 (t2 )]. Therefore, for all .n ≥ N, in view of (7.43) and (7.42), we have  (n)  −1 τ1 (s) − ε

τ1

.

   −1  − ≤ τ1 τ1−1 (s) − ε + Δ− ε /2 ≤ τ1 τ1 (s) − Δε /2

= s − Δ− ε /2    −1    + τ1(n) τ1−1 (s) + ε ≥ τ1 τ1−1 (s) + ε − Δ+ ε /2 ≥ τ1 τ1 (s) + Δε /2 = s + Δ+ ε /2 By the Intermediate Value Theorem, there exists some .t ∗ ∈ [τ1−1 (s)−ε, τ −1 (s)+ (n) ε] with .τ1 (t ∗ ) = s. In other words, we know     (n) −1  τ (s) − τ1−1 (s) = t ∗ − τ1−1 (s) < ε 1

.

∀n ≥ N

 (n) −1 This finishes our claim that . τ1 −→ τ1−1 uniformly on .[t1 , t2 ]. Let .τ2(n) (·), n ≥ 1, and .τ2 (·) be the exit time functions for commuters entering during the interval .[t1 , t2 ], corresponding to entering flow profiles .U (n) (·), n ≥ 1, and .U (·), respectively. Then for each .t ∈ [t1 , t2 ], we may state   (n)  (n) τ2 (t) = t + a U (n) (t) − U (n) (τ1 )−1 (t) + β,

.

n ≥ 1.

(7.44)

256

7 Dynamic Network Loading: Non-physical Queue Models

   τ2 (t) = t + a U (t) − U τ1−1 (t) + β

(7.45)



  (n) −1 Now we make the claim that .U (n) τ1 (t) −→ U τ1−1 (t) uniformly on .[t1 , t2 ]. Indeed, for any .ε > 0, there exists .N1 > 0 such that, if .n > N1 , we have   (n) U (t) − U (t) < ε/2

.

∀ t ∈ [t0 , tf ]

 (n) −1 Moreover, the functions . τ1 , . n ≥ 1, and .τ1−1 have a uniformly bounded range on .[t1 , t2 ], namely, .[t0 , t1 ]. By the Heine-Cantor theorem, .U (·) restricted to .[t0 , t1 ] is uniformly continuous, which means we can find .δ0 > 0 such that, for any .s1 , s2 ∈ [t0 , t1 ] with .|s1 − s2 | < δ0 , the following holds   U (s1 ) − U (s2 ) < ε/2

.

 (n) −1 By uniform convergence of . τ1 −→ τ1−1 , we may choose .N2 > 0 so that, for .n > N2 , we have   (n) −1  τ (t) − τ1−1 (t) < δ0 1

.

Thus we deduce that, given .n > max{N1 , N2 }, for any .t ∈ [t1 , t2 ], the following is true 



  (n)  (n) −1  τ1 .U (t) − U τ1−1 (t)    

 −1 −1    −1   ≤ U (n) τ1(n) (t) − U τ1(n) (t)  + U τ1(n) (t) − U τ1−1 (t)  < ε/2 + ε/2 = ε    −1 This shows the uniform convergence .U (n) τ1(n) (t) −→ U τ1−1 (t) on .[t1 , t2 ], and our claim is substantiated. As an immediate consequence of (7.44) (n) and (7.45), .τ2 converges to .τ2 uniformly on .[t1 , t2 ]. Part 3. We now proceed by induction as follows. Choose any .ν ≥ 2, and call . t˜ν+1 = inf τν(n) (tν ),

.

n

. tν+1 = t˜ν+1 − δ

where the constant .δ is the same as what was used in Part 2. Using the induction hypothesis that .τν(n) converges to .τν uniformly on .[tν−1 , tν ], we show the following uniform convergence on .[tν , tν+1 ]  .

τν(n)

−1

−→ τν−1

7.1 The Link Delay Model

257 (n)

The proof is similar to what has been done in Part 2. Now introduce .τν+1 (·), . n ≥ 1, and .τν+1 (·), which are the exit time functions corresponding to .U (n) (·), n ≥ 1, and .U (·), respectively; and they are restricted to the time interval .[tν , tν+1 ]. Similar to results (7.44) and (7.45), we have, for .t ∈ [tν , tν+1 ], the following   −1  (n) τν+1 (t) = t + a U (n) (t) − U (n) τν(n) (t) + β    τν+1 (t) = t + a U (t) − U τν−1 (t) + β

n ≥ 1.

.

(7.46) (7.47)

 (n) −1    −→ U τν−1 uniformly on It can be shown as before that .U (n) τν (n) .[tν , tν+1 ]. Therefore .τ ν+1 −→ τν+1 uniformly on .[tν , tν+1 ]. This finishes the induction.   Part 4. We now have obtained a sequence . [tν , tν+1 ] ν≥0 of intervals. On each interval .[tν , tν+1 ], the uniform convergence (n) τν+1 −→ τν+1

.

holds. Notice that, by construction, .tν+1 − tν ≥ β − δ > 0, .∀ν ≥ 1; therefore the interval .[t0 , tf ] can be covered by finitely many such intervals. As a consequence, we easily obtain the uniform convergence of .τ (n) (·) −→ τ (·) on the whole of .[t0 , tf ], where .τ (n) (·) corresponds to .U (n) (·), n ≥ 1, and .τ (·) corresponds to .U (·). Let .v (n) (·), .n ≥ 1, and .v(·) be the exit flows of the single arc and then define the cumulative exit vehicle counts  t  t . . .V (t) = v(s) ds, V (n) (t) = v (n) (s) ds, t0

t0

It then follows immediately from the relationships   V (t) = U τ −1 (t)

.

 −1 V (n) (t) = U (n) τ (n) (t)

that .V (n) converges uniformly to .V (·) on .[t0 , tf ]. Part 5. Consider a general network . A, V with a converging sequence .h(n) −→

|P | h in .Λ ⊂ L2 ([t0 , tf ]) . Define for .p ∈ P the following  (n) .Hp (t) (n)

≡

t

t0

h(n) p (s),

ds,

. Hp (t) =



t

hp (s) ds t0

Then the .Hp (·) converge uniformly to .Hp (·). For each arc .a ∈ A, where the notation employed has an obvious meaning, the cumulative entering vehicle

258

7 Dynamic Network Loading: Non-physical Queue Models (n)

count .Ua (·) is given by Ua(n) (t) =



.

Hp(n) (t) +

p

a  ∈I (a)

(n)

Va  (t)

In the above, the first summation is over all paths that use a as the first arc; and, in the second summation, .I(a) denotes the set of arcs immediately upstream from arc a. A simple mathematical induction with the results established in previous steps yields the uniform convergence Ua(n) (t) −→ Ua (t),

.

Va(n) (t) −→ Va (t),

τa(n) (t) −→ τa (t),

∀a ∈ A

where .τa (·)  is theexit time function of arc a. Thus, for  each  path .p ∈ P, the path delay .Dp ·, h(n) also converges uniformly to .Dp ·, h since it is a finite sum   of arc It remains to show that the effective delays obey .Ψp ·, h(n) −→  delays.  Ψp ·, h uniformly on .[t0 , tf ]. Recall   Ψp (t, h) = Dp (t, h) + F t + Dp (t, h) − TA

.

Notice that .F is uniformly continuous on .[t0 , tf ] by the Heine-Cantor theorem; this means, for any .ε > 0, there exists .σ > 0 such that whenever .|s1 − s2 | < σ , we have   F(s1 ) − F(s2 ) < ε/2

.

Moreover, by uniform convergence, there exits .N > 0 such that, for all .n > N, we have      Dp t, h(n) − Dp t, h  < min{σ, ε/2}

.

∀ t ∈ [t0 , tf ]

We then readily deduce that, given .n > N, the following holds for all .t ∈ [t0 , tf ]       Ψp t, h(n) − Ψp t, h        ≤ Dp t, h(n) − Dp t, h  



       + F t + Dp t, h(n) − TA − F t + Dp t, h − TA  .

< ε/2 + ε/2 = ε

7.2 The Vickrey Model

259

Final Argument Finally, uniform convergence on the compact interval .[t0 , tf ] implies convergence in the .L2 norm: 

tf

.

    2 Ψp t, h(n) − Ψp t, h dt −→ 0,

n −→ ∞,

p∈P

(7.48)

t0

    Summing up (7.48) over .p ∈ P, we obtain the convergence .Ψ ·, h(n) −→ Ψ ·, h  |P | in the Hilbert space . L2 ([t0 , tf ]) .  In addition to the strong continuity established in the previous theorem, sometimes a different notion of continuity is used, for example, in the proof of solution existence (assumption A3 in Chap. 5). This is recalled below. A3. For any sequence of departure rate vectors .{h(n) }n≥1 ⊂ Λ that are uniformly bounded by a positive constant and converge weakly to .h∗ ∈ Λ, the corresponding effective path delays .Ψp (t, hn ) converge to .Ψp (t, h∗ ) uniformly for all .p ∈ P and .t ∈ [t0 , tf ] as .n → +∞. Theorem 7.3 Under the same conditions as in Theorem 7.2, the effective path delay  |P | , .h → Ψ (·, h) is continuous in the sense of .A3. operator .Ψ : Λ → L2 [t0 , tf ] Proof The proof deviates from that of Theorem 7.2 only in Part 1. In particular, we focus on a single link and consider a uniformly bounded sequence of entering flows (n) , .ν ≥ 1 that converge to u weakly. We define .u . U (n) (t) =



t

.

u(n) (s) ds,

t0

. U =



t

u(s) ds

∀t ∈ [t0 , tf ]

t0

Then one immediately has that .U (n) (·) converges to .U (·) uniformly on .[t0 , tf ]. The rest of the proof is the same as the proof of Theorem 7.2. 

7.2 The Vickrey Model The Vickrey model (VM) is one of the most commonly employed link models in the literature of traffic flow. It was originally presented in Vickrey (1969) and discussed subsequently, for example, by Drissi-Ka¨touni and Hameda-Benchekroun (1992), Heydecker and Addison (1996), Kuwahara and Akamatsu (1996), and Li et al. (2000). The Vickrey model is based on the assumption that the queue has negligible size and that the travel time on the link consists of a free flow travel time plus a congestion-related queuing time. A popular mathematical form of the model is an ordinary differential equation with discontinuous dependence on the state variable. Such an irregularity leads to several theoretical difficulties: a classical solution may no longer exist; and it is widely suspected that such an ODE does not admit an

260

7 Dynamic Network Loading: Non-physical Queue Models

explicit solution representation. For a general review on ODEs with discontinuous state dependence, the reader is referred to Filippov (1988) and Stewart (1990).

7.2.1 Ordinary Differential Equation Formulation The Vickrey model is based on two primary assumptions: (1) the vehicles have negligible sizes, and therefore queues do not occupy any physical space; and (2) the link traversal time consists of a fixed travel time plus a congestion-related queuing time. This model takes various mathematical forms in the literature. One of the most recognized formulations is discussed by Kuwahara and Akamatsu (1996) and Nie and Zhang (2005), which will be presented below. Let us begin by introducing the following notations. .

u(t) :

the link entering flow

w(t) :

the link exiting flow

q(t) :

the queue size

M :

the flow capacity of the bottleneck

t0 :

the free flow travel time

λ(t) :

the link traversal time of drivers entering the link at t

Consider a time horizon .[0, T ], with any .T > 0. The link flow dynamic can be described as follows. (1) Vehicles entering the link move with free flow speed before arriving at the exit, which is a bottleneck; (2) a queue with zero physical size (point queue) forms at the exit if the arriving flow exceeds the bottleneck capacity. The rate at which vehicles are released from the queue is described by  w(t) =

.

  min u(t − t0 ), M if q(t) = 0 if q(t) > 0

M

t0 ≤ t ≤ T

(7.49)

The rule for releasing vehicles from the queue is straightforward: vehicles are released at the maximum rate allowed by the link capacity and vehicles supplied to the queue. Once the flow arriving at the queue .u(t − t0 ) and the flow leaving the queue .w(t) are determined, the change rate of the queue can be described by dq(t) = . dt



0

if 0 ≤ t < t0

u(t − t0 ) − w(t) if t0 ≤ t ≤ T

(7.50)

By (7.50), it is assumed that the link is empty at .t = 0. Notice that if the queue is nonempty, then the time it takes to traverse this queue is proportional to .q(t).

7.2 The Vickrey Model

261

Therefore, for a driver entering the link at t, the link traversal time can be expressed as λ(t) = t0 +

.

q(t + t0 ) M

0 ≤ t ≤ T

(7.51)

The Vickrey model is described collectively by (7.49)–(7.51). Notice that (7.49) and (7.50) can be combined to form a single ordinary differential equation   dq(t) min u(t − t0 ), M} if q(t) = 0 = u(t − t0 ) − . dt M if q(t) > 0

t0 ≤ t ≤ T (7.52)

The case when .0 ≤ t ≤ t0 is trivial since the queue remains empty. One important observation is that the right-hand side of ODE (7.52) is discontinuous in the unknown variable .q(t). To see this, consider the case where .q(t) > 0 and .u(t − t0 ) < M. As .q(t) approaches zero, the exit flow .w(t) will jump from M down to .u(t − t0 ). Such a discontinuous dependence implies that the ODE (7.52) does not have any classical solution, i.e., solutions that are locally continuously differentiable. Instead, one may consider the solution in the integral sense, i.e., solution that satisfies the differential equation at almost every .t ∈ [0, T ] or, more generally, solution in the distributional (weak) sense. Due to the irregularity inherent in the ODE (7.52), a direct analysis of the aforementioned Vickrey’s model in continuous time is difficult: one needs to rely on the mathematical tools for discontinuous ODEs, as presented in Filippov (1988) and Stewart (1990). However, a few modifications of the Vickrey model have been proposed in the literature that allow further analysis of the dynamics in continuous time. In this section, we will briefly review these models and discuss their merits.

7.2.1.1

The Linear Complementarity System Formulation

In the study of a DUE problem with a single bottleneck, Pang et al. (2011) employ a time-dependent linear complementarity system (LCS) as an alternative to the ODE (7.52). The LSC can be explicitly written as .

dq(t) = s(t) + u(t − t0 ) − M dt

(7.53)

0 ≤ s(t) ⊥ q(t) ≥ 0

(7.54)

.

Here .s(t) is a time-dependent slack variable, and the rest of the notations are same as before. System (7.53)–(7.54) does not admit an explicit ODE formulation but has

262

7 Dynamic Network Loading: Non-physical Queue Models

the implicit ODE form .

  dq(t) + M − u(t − t0 ) = 0 min q(t), dt

(7.55)

The LCS model explicitly imposes the non-negativity of the queue size .q(t). As mentioned in Ban et al. (2011), this model deviates from Vickrey’s original model in the case when .q(t) = 0. Indeed, the Vickrey model implies in this case that   dq(t) dq(t) . dt = max u(t − t0 ) − M, 0 , while (7.55) merely implies . dt ≥ u(t − t0 ) − M. As far as finite-difference schemes for (7.53)–(7.54) are concerned, a forward (explicit) scheme is not well-defined. Namely, given the state variables at the current time step, the state variables at the next time step are underdetermined. See Ban et al. (2011) for more detail. However, a backward (implicit) scheme is well-defined and coincides with the numerical scheme proposed in Nie and Zhang (2005). In addition, the non-negativity of the queue length is guaranteed under such a scheme because of (7.54).

7.2.1.2

The α-Model

Ban et al. (2011) proposed a novel approach which transformed the above LCS into an explicit ODE. This was done by introducing a parameter α  1 and by writing .

  dq(t) = max u(t − t0 ) − M, −α q(t) dt

(7.56)

Hereafter, we refer to the above equation as the α-model, a name suggested by Ban et al. (2011). It has been shown by the same authors that the continuous-time solution to (7.56) guarantees the non-negativity of both queue size and exit flow, through a non-trivial argument. Moreover, as α → +∞, the solution of ODE (7.56) converges to the solution of the LCS above. However, we note that the α-model also deviates from the Vickrey model: if q(t) = 0 and u(t − t0 ) − M + αq(t) < 0, then .

dq(t) = − α q(t) > u(t − t0 ) − M dt

while both the Vickrey model and the LCS stipulate that dq(t) dt = u(t − t0 ) − M. Regarding numerical performance of the model, it was demonstrated in Ban et al. (2011) that an explicit (forward) discretization scheme for the α-model may result in negative queues. Moreover, the stability of the explicit scheme is conditional. On the other hand, the implicit (backward) scheme ensures the non-negativity of the queue and the flow while approximating the LCS model well if the value of α is appropriately selected.

7.2 The Vickrey Model

263

The remaining question for the α-model is how to appropriately select the numerical values of α to ensure the proper functioning of this model. In a dynamic and complex network setting where the flows on each link are highly variable, the selection of α is crucial. Furthermore, it would be interesting to further explore the physical meaning and the underlying modeling implications of the parameter α.

7.2.1.3

The ε-Model

Models that rely on point queue-type dynamics appear also in the modeling of continuous supply chain networks. In particular, Armbruster et al. (2006) and Fügenschuh et al. (2008) employed the following link dynamic, which is very similar to that of the Vickrey model.    d min u(t), M if q(t) = 0 q(t) = u(t) − . dt M if q(t) = 0

(7.57)

Notice that (7.57) implies that the queue of goods/products is located at the entrance of each link (processor), instead of at the exit. In order to smooth out the discontinuity in the right-hand side of ODE (7.57), a smoothing parameter .0 < ε  1 is used so that (7.57) becomes   q(t) d q(t) = u(t) − min M, . dt ε

(7.58)

Hereafter, we call (7.58) the .ε-model. We observe that the ODE (7.58) is stiff whenever .0 < q(t) < εM. A stiffness condition needs to be imposed for the explicit scheme to ensure stability. One such is Δt ≤ ε

.

(7.59)

where .Δt denotes the time step size of the discretization. The .ε-model and the .αmodel share a similar mathematical structure. Such an assertion is supported by the following slightly different presentation of (7.58): .



 1 d q(t) = max u(t) − M, εu(t) − q(t) dt ε

(7.60)

where .ε can be treated as .1/α. In regimes where .α is large or .ε is small, the systems (7.60) and (7.56) tend to exhibit the same qualitative behavior. The .ε-model has an explicit ODE representation. It is a practical alternative to (7.57), as was verified extensively in Armbruster et al. (2006) against deterministic discrete event simulations. However, one would expect that the solution quality will deteriorate for larger values of .ε. One such example is presented by

264

7 Dynamic Network Loading: Non-physical Queue Models

Han et al. (2012). In view of (7.59), there is a trade-off among numerical stability, computational efficiency, and solution quality. We note here that both the .α-model and the .ε-model rely on an appropriate choice of an exogenous parameter, which is influenced by the numerical scales of variables such as .t, q(t), and .u(t). Such a fact could potentially raise the issue of scalability of these models. In other words, a case-dependent strategy for choosing the parameters is necessary to ensure the overall performance of these models. Each of the modifications of the Vickrey model we have presented so far has its own merits in analysis and application. The alternative models approximate the Vickrey model under certain conditions and the corresponding numerical methods manage to fix the problems arising from the time discretization of the original Vickrey’s ODE. In addition, the approximate formulations provide valuable analytical and computational insights into the Vickrey model.

7.2.2 Partial Differential Equation Formulation Han et al. (2013a) propose a novel reformulation of the Vickrey model as a partial differential equation (PDE). Such an approach provides unique insights and analytical tools that are unavailable through theoretical or numerical study of the ODE (7.52). Sections 7.2.2 and 7.2.3 recap the derivation and analysis of the equivalent PDE formulation and highlight several important results, including a closed-form solution of the ODE (7.52). Recall the Vickrey model in the form of an ordinary differential equation:   dq(t) = u t − t0 − w(t), q(t0 ) = 0 t 0 ≤ t ≤ T + t0 dt ⎧ ⎪ 0 0 ≤ t ≤ t0 ⎪ ⎨   .w(t) = min{u t − t0 , M} if q(t) = 0 ⎪ t0 < t ≤ T + t 0 ⎪ ⎩ M if q(t) > 0 .

λ(t) =

.

q(t + t0 ) + t0 M

0 ≤ t ≤ T

(7.61)

(7.62)

(7.63)

where q(t) denotes the queue size, u(t) denotes the link inflow, w(t) denotes the link exit flow, and λ(t) denotes the time taken to traverse the link when the time of entrance is t, t0 is the fixed free flow travel time, and M is the flow capacity of the bottleneck. The key ingredient of a PDE reformulation of system (7.61)–(7.63) is a virtual spatial dimension x ∈ [0, L], where L is the link length. We introduce free flow speed v0 such that L = v0 t0 . Similar to the LWR model, the Vickrey model can be

7.2 The Vickrey Model

265

described based on the conservation of vehicles ∂t ρ(t, x) + ∂x f (t, x) = 0

(t, x) ∈ [0, T ] × [0, L]

.

(7.64)

where ρ(t, x) and f (t, x) denote, respectively, the vehicle density and flow at a point in the temporal-spatial domain. Notice that a PDE of the form (7.64) is always true for any dynamics prescribed by the conservation of mass. What distinguishes the Vickrey model from the LWR model is the density-flow relationship. In the classical LWR model, flow is a function of density only, which is given by the fundamental diagram, while in the Vickrey model, the flow depends not only on density but also on the spatial variable x, due to the presence of a bottleneck at the link exit x = L. Indeed, the flow f (t, x) can be expressed via a flux function φ

. .f (t, x) = φ x, ρ(t, x) =



v0 ρ(t, x)   min M, v0 ρ(t, x)

if x ∈ [0, L) if x = L

(7.65)

Another way of interpreting the flux function φ is as follows. We introduce the xdependent capacity function M(x), which is defined as  . .M(x) =

+∞ if x ∈ [0, L) M

if x = L

(7.66)

The definition above is quite straightforward: in the free flow phase x ∈ [0, L), vehicles always travel at a constant speed no matter how large the flow may be, implying that the flow capacity is infinite, whereas at the exit x = L, a finite capacity constraint applies. The flux function φ can be alternatively defined as

  φ x, ρ(t, x) = min M(x), v0 ρ(t, x) ,

.

(7.67)

which explicitly imposes the flow capacity constraint everywhere. On can easily check that the two definitions (7.65) and (7.67) of φ are equivalent. Summarizing what has been discussed, we present the conservation law formulation of the Vickrey model ⎧   ⎨∂ ρ(t, x) + ∂ min M(x), v ρ(t, x) = 0 (t, x) ∈ [0, T ] × [0, L] t x 0 . ⎩v0 ρ(t, 0) = u(t) t ∈ [0, T ] (7.68) where M(x) is given by (7.66). u(t) is the link inflow. Remark 7.2 Equation (7.68) is a scalar conservation law with x-dependent flux function. A general review on mathematical properties of such PDEs can be found in Evans (1995). Notice that due to the discontinuous dependence of the flux function on x, the solution to PDE (7.68) can only be considered in the distributional sense. In

266

7 Dynamic Network Loading: Non-physical Queue Models

other words ρ(t, x) may contain δ-distribution, which is a mathematical abstraction of the notion of “point queue.”1 A Hamilton-Jacobi equation can be obtained by integrating the conservation law (7.68). Let us introduce the cumulative vehicle counts . .N (t, x) =



t

f (s, x) ds,

. U (t) =



0

t

t ∈ [0, T ], x ∈ [0, L]

u(s) ds 0

(7.69) N (t, x) measures the number of vehicles that have passed location x by time t. U (·) ≡ N(·, 0) is the cumulative entering vehicle count. N(·, ·) satisfies the following Hamilton-Jacobi equation: ⎧   ⎨∂ N (t, x) − min M(x), −v ∂ N(t, x) = 0 t 0 x . ⎩N (t, 0) = U (t)

(t, x) ∈ [0, T ] × [0, L] t ∈ [0, T ] (7.70)

It is important to note that the Hamiltonian in the H-J equation (7.70)   H(x, p) = min M(x), −v0 p

.

has a discontinuous x-dependence, due to the discontinuous function M. Experience with PDEs suggests that the solution of (7.70) exists only in the sense of distribution. In the next section, we will present a variational method for solving such an equation.

7.2.3 Closed-Form Solutions of the Vickrey Model We employ the variational method, namely, the Lax-Hopf formula mentioned in Chap. 2, in search of a solution representation of the H-J equation (7.70). For the completeness of our presentation, we briefly review the general Lax-Hopf formula for a Hamilton-Jacobi equation with x-independent Hamiltonian. Consider the following Cauchy problem (initial-value problem). ⎧

⎨∂ N (t, x) + H ∂ N (t, x) = 0 t x . ⎩N (0, x) = g(x)

(t, x) ∈ (0, ∞) × R x∈R

(7.71)

1 For this reason, many scholars refer to the Vickrey model as the point queue model (PQM). However, as we commented in the beginning of this chapter, the name PQM was first suggested by Daganzo (1994) to describe the link delay model (LDM) proposed in Friesz et al. (1993).

7.2 The Vickrey Model

267

Theorem 7.4 (The Lax-Hopf Formula) Suppose .H(·) is continuous and convex, g(·) : R → R is Lipschitz continuous. Then

.

    x−y + g(y) N (t, x) = inf t · L y∈R t

.

(7.72)

is the unique viscosity solution to the initial-value problem (7.71), where .L is the Legendre transformation of .H   L(q) = inf H(p) − qp

.

p

(7.73) 

Proof See Evans (1995).

Remark 7.3 We note that formula (7.72) does not immediately apply to Eq. (7.70). This is because (1) the Hamiltonian .H(·) in (7.71) is x-independent; (2) (7.71) contains an initial condition instead of a boundary condition; and (3) in order to apply an (7.72), the initial datum .g(·) is required to be Lipschitz continuous, which d does not hold for .U (·) in (7.70), since . dt U (t) = u(t) is unbounded. In order to overcome the aforementioned difficulties, we consider a flow-based conservation law whose primary variable is the flow .f (t, x):   ⎧ 1 ⎨∂ f (t, x) + ∂ f (t, x) = 0 (t, x) ∈ [0, T ] × [0, L) x t v0 . ⎩ f (t, 0) = u(t) t ∈ [0, T ]

(7.74)

Notice that in the PDE above, the domain of x is .[0, L) without the right boundary point. The PDE in (7.74) is obviously equivalent to (7.64) as long as the boundary point .x = L is excluded. Next we apply the formula from Theorem 7.4 to the following Hamilton-Jacobi equation, which is equivalent to (7.74). ⎧ ⎨∂ N(t, x) + 1 ∂ N(t, x) = 0 t x v0 . ⎩ N (t, 0) = U (t)

(t, x) ∈ [0, T ] × [0, L)

(7.75)

t ∈ [0, T ]

where .N (t, 0) and .U (t) are defined in (7.69). Remark 7.4 Notice that the system (7.75) can be interpreted as an initial-value problem by switching the roles of t and x. This technique enables us to apply Theorem 7.4. The next lemma is an application of the Lax-Hopf formula to (7.75), in the case where .U (·) is Lipschitz continuous.

268

7 Dynamic Network Loading: Non-physical Queue Models

Lemma 7.2 Assume that .U (·) is Lipschitz continuous with Lipschitz constant M. Then the viscosity solution to (7.75) is given by the following. N (t, x) =

.

min

0≤τ ≤t− vx

0

    x U (τ ) − Mτ + M · t − v0

(t, x) ∈ [0, T ] × [0, L) (7.76)

Proof The proof follows that in Han et al. (2013a). By the hypothesis, the flow f (t, x) = ∂t N (t, x) is uniformly bounded by M. Instantiating (7.71) with (7.75), we have

.

1 p v0

H(p) =

.

p ∈ [0, M]

The Legendre transformation becomes ⎧ ⎨0 .L(q) = ⎩ q−

1 v0

q ≤

M

q >

1 v0 1 v0

(7.77)

Applying formula (7.72) with switched t and x, we readily deduce     t −τ x·L + U (τ ) .N(t, x) = inf τ ∈R x

(7.78)

In view of (7.77), two cases may arise for (7.78): (i)

.

t−τ x

≤

1 v0 .

Then (7.78) becomes x inf x U (τ ) = U t − v0 τ ≥t− v

N(t, x) =

.

(7.79)

0

(ii)

The last equality is due to the monotonicity of .U (·). 1 t−τ x > v0 . Then (7.78) becomes

.

N(t, x) =

.

 inf

0≤τ W (τt∗ ). Then the left-continuity of .U (· − t0 ) at ∗ ∗ ∗ .τt implies there exists .δ > 0 such that .U (τ − t0 ) > W (τ ), ∀τ ∈ (τt − δ, τt ]. By (7.84), we deduce 

W (τt∗ ) − W (τ ) =

τt∗

.

τ

M ds = M(τt∗ − τ ) ⇒ W (τ ) − Mτ = Δt

∀τ ∈ (τt∗ − δ, τt∗ ]

This yields a contradiction to (7.87); thereby, the claim is substantiated. Finally, the fact that .U (τt∗ − t0 ) = W (τt∗ ) leads to .

min

t0 ≤τ ≤t



U (τ − t0 ) − Mτ



≤ U (τt∗ − t0 ) − Mτt∗ = Δt =

min

t0 ≤τ ≤t



W (τ ) − Mτ



proves (7.86). Next, by Lipschitz continuity of .W (·), it follows that .

min

t0 ≤τ ≤t



W (τ ) − Mτ



= W (t) − M t

(7.88)

Therefore, we deduce from (7.88) and (7.86) that W (t) =

.

=

min

t0 ≤τ ≤t

min

  W (τ ) − Mτ + M t =

0≤τ ≤t−t0



min

t0 ≤τ ≤t

U (τ ) − M τ } + M(t − t0 )



 U (τ − t0 ) − M τ + M t 

Remark 7.8 Although the above theorem is a consequence of the PDE formulation, (7.85) itself is independent of any spatial variable or any assumption related to the PDE. Such a formula remains true even if .t0 = 0.

272

7 Dynamic Network Loading: Non-physical Queue Models

Theorem 7.5 combined with Lemma 7.2 provides a closed-form solution to the H-J equations (7.70). The fact that the formula for .x ∈ [0, L) can be extended . to include .x = L is non-trivial. Later we will show that the quantity .q(t) = U (t − t0 ) − W (t) where .W (t) is given by (7.85) indeed solves Vickrey’s ODE in the integral sense. The impact of Eq. (7.85) is threefold. (1) The solution .W (t) is given in closed-form, providing opportunity for further mathematical analysis that is relatively difficult to conduct based on the ODE. (2) The exit vehicle count .W (·) can be computed directly from the entering vehicle count without any intermediate computation of the queue size. (3) The solution given by (7.85) works for very general flow profiles such as the ones in Definition 7.1. The next proposition establishes several mathematical properties of the resulting continuous-time solution that provides reasonable physical realism. Proposition 7.1 (Mathematical Properties of the Solution) Given .U (·) [0, T ] → R+ defined in Definition 7.1, the following three statements hold:

:

(1) The exiting vehicle count W (t) =

.

  U (τ ) − Mτ + M(t − t0 )

min

0≤τ ≤t−t0

t ∈ [t0 , T + t0 ]

(7.89)

is non-negative, non-decreasing, and Lipschitz continuous with Lipschitz constant M. (2) The queue size q(t) = U (t − t0 ) − W (t) = U (t − t0 ) − M(t − t0 ) −

.

min

0≤τ ≤t−t0

  U (τ ) − Mτ (7.90)

for all .t ∈ [t0 , T + t0 ] is non-negative. In addition, if .U (·) is absolutely continuous, then so is .q(·). (3) (First-in-first-out) Let the link traversal time of a vehicle entering at time t be q(t + t0 ) . λ(t) = t0 + M

.

t ∈ [0, T ]

(7.91)

where .q(t) satisfies (7.90). Then for any .0 ≤ t1 < t2 ≤ T t1 + λ(t1 ) ≤ t2 + λ(t2 )

.

(7.92)

The equality of (7.92) holds if and only if .U (t1 ) = U (t2 ) and .w(t) ≡ M .∀t ∈ [t0 + t1 , t0 + t2 ].

7.2 The Vickrey Model

273

Proof We follow Han et al. (2013a). (1) We start by showing that .W (·) is non-decreasing. Fix any .t1 , t2 such that .t0 ≤ t1 ≤ t2 ≤ T + t0 ; we have W (t2 ) − W (t1 ) =

.



min

0≤τ ≤t2 −t0

 U (τ ) − Mτ −

min



0≤τ ≤t1 −t0

U (τ ) − Mτ



+ M(t2 − t1 ) ≥ − M(t2 − t1 ) + M(t2 − t1 ) = 0 Then the non-negativity of .W (·) follows from monotonicity and the fact that W (t0 ) = 0. In order to show that the right-hand side of (7.89) is Lipschitz continuous, we deduce, for any .t0 ≤ t1 ≤ t2 ≤ T + t0 , that

.

0 ≤ W (t2 ) − W (t1 )   = min U (τ ) − Mτ −

.

0≤τ ≤t2 −t0

min

0≤τ ≤t1 −t0

  U (τ ) − Mτ + M(t2 − t1 )

≤ M(t2 − t1 ) (2) The non-negativity of .q(·) is obvious from (7.90). Notice that .q(·) is the difference of .U (· − t0 ) and a Lipschitz (hence absolutely) continuous function .W (·). Therefore, .q(·) will be absolutely continuous provided that .U (·) is absolutely continuous. (3) Using (7.90) and (7.91), we equivalently write (7.92) as U (t1 ) − min

.

0≤τ ≤t1

    U (τ ) − Mτ ≤ U (t2 ) − min U (τ ) − Mτ 0≤τ ≤t2

which is always true since U (t1 ) ≤ U (t2 )

and

.

min

0≤τ ≤t1



   U (τ ) − Mτ ≥ min U (τ ) − Mτ 0≤τ ≤t2

In order for the equality in (7.92) to hold, one must have U (t1 ) = U (t2 )

and

.

min

0≤τ ≤t1



U (τ ) − M τ



=

min

0≤τ ≤t2



U (τ ) − M τ



The second identity implies the following W (t0 + t2 ) − W (t0 + t1 ) = M(t2 − t1 )

.

This means that .w(t) ≡ M .∀t ∈ [t0 + t1 , t0 + t2 ].



274

7 Dynamic Network Loading: Non-physical Queue Models

The sufficient and necessary condition provided in (3) implies that if two drivers entering the link at different times were to exit the link at the same time, then (a) there must be no other cars between them and (b) the driver that enters first must wait in a non-empty queue until the second driver catches up with him/her. Let us now return to the original ODE model based on (7.61)–(7.62), which can be compactly written as the following ODE with discontinuous right-hand side  dq(t) min{u(t − t0 ), M} if q(t) = 0 = u(t − t0 ) − . dt M if q(t) > 0

t ∈ [t0 , T + t0 ] (7.93)

We assert that the solution derived from the PDE approach indeed solves the ODE above. To make our assertion precise, let us define a proper solution class for the ODE. Definition 7.2 (Strong Solution of the ODE) Given an integrable function .u(·) : [0, T ] → R+ as in (7.93), the strong solution (also called the integral solution) to the ODE (7.93) is defined to be an absolutely continuous function .q(·) : [t0 , T + t0 ] → R+ with .q(0) = 0, which satisfies (7.93) at almost every .t ∈ [t0 , T + t0 ]. Theorem 7.6 The explicit formula for the queue size .q(t) given in Proposition 7.1 q(t) = U (t − t0 ) − M (t − t0 ) −

.

 min

0≤τ ≤t−t0

U (τ ) − Mτ



t ∈ [t0 , T + t0 ] (7.94)

is the strong solution to the ODE (7.93) in the sense of Definition 7.2. Here the absolutely continuous function .U (·) is defined by . U (t) =



t

u(s) ds

.

t ∈ [0, T ]

0

Proof We follow Han et al. (2013a). First we define a Lipschitz continuous function with Lipschitz constant M . W (t) =

.

min

0≤τ ≤t−t0

  U (τ ) − Mτ + M(t − t0 )

t ∈ [t0 , T + t0 ]

(7.95)

Notice that by (7.95), we have .U (· − t0 ) ≥ W (·). Equation (7.94) becomes q(t) = U (t − t0 ) − W (t). It then follows immediately that .q(·) is absolutely continuous and

.

.

d d q(t) = u(t − t0 ) − W (t) dt dt

for almost every t ∈ [t0 , T + t0 ]

7.2 The Vickrey Model

275

Notice that the above differentiations are well-defined due to absolute continuity. It remains to show that, for almost every .t ∈ [t0 , T + t0 ], d . W (t) = dt

 min{u(t − t0 ), M} if q(t) = 0

(7.96)

if q(t) > 0

M

We fix any .t ∈ (t0 , T + t0 ] \ Ω, where .Ω is the set of points where .U (· − t0 ) or W (·) is not differentiable. Besides, .Ω has zero measure. Two cases may arise.

.

(i) .q(t) = 0. Then .U (t − t0 ) = W (t). By definition (7.95), we have U (t − t0 ) − M(t − t0 ) =

.

min

0≤τ ≤t−t0

  U (τ ) − Mτ

(7.97)



d U (t − t0 ) − M(t − t0 ) ≤ 0. Otherwise, consider .ε > 0 small We claim that . dt d enough that . dt U (t − t0 ) > M + ε. Then there exists .δ > 0 such that

.

U (t − t0 ) − U (t1 − t0 ) > M +ε t − t1

∀ t1 ∈ [t − δ, t),

contradicting (7.97). Thereby the claim is substantiated. d d W (t) = dt U (t − t0 ). Indeed, consider any sequence Next, we show that . dt .tn > t ∀n ∈ N such that .tn → t. Recalling .W (·) ≤ U (· − t0 ), we have .

U (tn − t0 ) − U (t − t0 ) W (tn ) − W (t) ≤ tn − t tn − t

∀ n∈N

d d Thus . dt W (t) ≤ dt U (t − t0 ). We now apply the same argument to any sequence d d .tν < t ∀ν ∈ N such that .tν → t and get . dt W (t) ≥ dt U (t − t0 ). We conclude d d that . dt W (t) = dt U (t − t0 ), as promised. d d W (t) = dt U (t − t0 ) = u(t − t0 ). The Lipschitz continuity We have that . dt of .W (·) indicates that .u(t − t0 ) ≤ M. This shows the first case of (7.96). (ii) .q(t) > 0. In this case, we have

U (t − t0 ) − M(t − t0 ) >

.

 min

0≤τ ≤t−t0

U (τ ) − Mτ



By continuity of .U (·), there exists a right neighborhood .Nt+ of t such that U (tˆ − t0 ) − M(tˆ − t0 ) >

.

min

0≤τ ≤t−t0

  U (τ ) − Mτ

∀tˆ ∈ Nt+

276

7 Dynamic Network Loading: Non-physical Queue Models

This implies that W (tˆ ) =

.

=

min

0≤τ ≤tˆ−t0

min

0≤τ ≤t−t0



 U (τ ) − M τ + M(tˆ − t0 )

  U (τ ) − Mτ + M(tˆ − t0 )

∀ tˆ ∈ Nt+

Therefore, we take any sequence .tn ∈ Nt+ ∀n ∈ N with .tn → t, and deduce .

W (tn ) − W (t) d W (t) = lim = M tn →t dt tn − t 

This shows the second case of (7.96).

7.2.4 The Generalized Vickrey Model The Vickrey model is expressed as ODE (7.52), which implicitly requires that the inflow .u(·) is an integrable function. Equivalently, the entering vehicle count .U (·) should be absolutely continuous. This assumption, however, might be restrictive for the analysis of dynamic traffic assignment (DTA) problems. As demonstrated by Bressan and Han (2011) and Han et al. (2013b), DTA models are more easily analyzed in a mathematical framework if the embedded dynamic network loading models can treat network flow that exists in the distributional sense. Notice that the formulae in Proposition 7.1 do not require continuity of .U (·). In fact, by Definition 7.1, .U (·) only needs to be non-decreasing and left-continuous. This suggests a way of extending the Vickrey model to a more general setting without invoking the ODE. In the following definition, we present the generalized Vickrey model (GVM) for a link that is initially empty. Definition 7.3 (GVM with Zero Initial Condition) The generalized Vickrey model is defined in terms of cumulative vehicle counts. Assume the link is empty at .t = 0. Given the entering vehicle count .U (·) : [0, T ] → R+ , which is nondecreasing and left-continuous, the exiting vehicle count .W (·) : [0, T ] → R+ is then given by W (t) =

.

⎧ ⎨0 ⎩ min

0≤τ ≤t−t0

  U (τ ) − Mτ + M(t − t0 )

t ∈ [0, t0 ) t ∈ [t0 , T + t0 ]

(7.98)

The arc traversal time function .λ(·) : [0, T ] → [0, +∞) is given by     1 · U (t) − Mt − min U (τ ) − Mτ .λ(t) = t0 + 0≤τ ≤t M

t ∈ [0, T ]

(7.99)

7.2 The Vickrey Model

277

. d In the case where .U (·) is absolutely continuous and . dt U (t) = u(t), the generalized Vickrey model reduces to the original ODE-based system (7.61)–(7.63). In the GVM, the link inflow is no longer assumed to be a classical function. Rather, it is considered in the distributional sense.

7.2.4.1

Initial-Boundary Value Problem

We consider the initial-boundary value problem for GVM. Assume that, at time t = 0, we are given an initial condition .ρ0 (x) ∈ L1 [0, L] for the car density on the link and .q0 ∈ R+ for the queue size. Namely,

.

ρ(0, x) = ρ0 (x),

.

x ∈ [0, L),

q(0) = q0

(7.100)

This problem is mathematically expressed as the following H-J equation with initial and boundary conditions   ⎧ ⎪ ∂ N (t, x) − min M(x), −v ∂ N(t, x) = 0 ⎪ t 0 x ⎨ . N (t, 0) = U (t), ⎪ ⎪ L ⎩ N (0, x) = x ρ0 (y) dy + q0 ,

(t, x) ∈ [0, T ] × [0, L] t ∈ [0, T ] x ∈ [0, L] (7.101)

Proposition 7.2 (GVM with Initial-Boundary Conditions) Given the initial conditions (7.100) and boundary condition .U (·), the cumulative exiting vehicle count .W (·) and the arc traversal time function .λ(·) are given by W (t) =

.

1 .λ(t) = t0 + M

  min U (τ ) − Mτ + M t

t ∈ [0, T + t0 ]

0≤τ ≤t

 U (t + t0 ) − M(t + t0 ) −

min

0≤τ ≤t+t0

  U (τ ) − Mτ

(7.102)  t ∈ [0, T ] (7.103)

where .U (·) is defined as ⎧ ⎪ 0 ⎪ ⎪  L ⎪ ⎪ ⎨ . q0 + ρ0 (y) dy .U (t) = L−tv0 ⎪  ⎪ L ⎪ ⎪ ⎪ ⎩q0 + ρ0 (y) dy + U (t − t0 )

t = 0 t ∈ (0, t0 ]

(7.104)

t ∈ (t0 , T + t0 ]

0

Proof Let .U˜ (·) : [0, T +t0 ] → R+ measure the cumulative number of vehicles that have arrived at the queue. Clearly .U˜ (0) = q0 . In addition, during the time interval

278

7 Dynamic Network Loading: Non-physical Queue Models

(0, t0 ], the cars initially on the link will arrive at the queue at a rate given by

.

f (t, L−) = v0 ρ0 (L − v0 t)

t ∈ (0, t0 ].

.

The above identity follows from a simple calculation using the method of characteristics, since the dynamic on .x ∈ [0, L) is governed by a linear advection equation. We have for .t ∈ (0, t0 ] U˜ (t) = q0 +



t

.

 v0 ρ0 (L − v0 s) ds = q0 +

L

ρ0 (y) dy

0

L−v0 t

Finally, for the time interval .(t0 , T +t0 ], the flow arriving at the queue is determined by .U (· − t0 ). We thus have ˜ (t) = q0 + .U



L

ρ0 (y) dy + U (t − t0 )

t ∈ (t0 , T + t0 ]

0

Notice that we need to pick the left-continuous version of .U˜ (·), which is given by (7.104). Identities (7.102) and (7.103) follow immediately from the definition of GVM.  Remark 7.9 Notice that the left-continuity dictates that .U (0) = 0. The formulae would not be correct if one uses .U˜ (·) instead. Example 7.1 (Continuous-Time Solutions of the GVM) We assume the free flow time .t0 = 0 for simplicity. Consider the following inflow profile u(t) = cos(t − π ) + 1,

.

U (t) = sin(t − π ) + t

t ∈ [0, 10]

(7.105)

  The key step of computing the GVM is to find . min U (τ ) − Mτ . In order 0≤τ ≤t

to understand such a quantity, we provide four different cases with varying flow capacities .M = 0.5, 1, 1.5, 2. The four cases are depicted in Figs. 7.2, 7.3, 7.4, and 7.5, respectively. (i) .M = 0.5. Function .U (τ ) − Mτ has a global minimizer .τ ∗ = π/3. .U (τ ∗ ) − Mτ ∗ = −0.3424. We have    U (t) − Mt if 0 ≤ t ≤ π/3 U (τ ) − Mτ = . min 0≤τ ≤t −0.3424 if π/3 < t ≤ 10 The solution to the ODE (7.93) reads  q(t) =

.

0

if 0 ≤ t ≤ π/3

sin(t − π ) + 0.5t + 0.3424

if π/3 < t ≤ 10

7.2 The Vickrey Model

279

12

U(t)=sin(t−π)+t M=0.5

10

Global Minimizer 8

6

4

2

0

0

2

4

6

8

10

6

8

10

t Fig. 7.2 Case (i): .M = 0.5 12

U(t)=sin(t−π)+t M=1

10

Global Minimizers 8

6

4

2

0

0

2

4

t Fig. 7.3 Case (ii): .M = 1

280

7 Dynamic Network Loading: Non-physical Queue Models 12

U(t)=sin(t−π)+t M=1.5

10

Local Minimizer Global Minimizer

8

p 6

4

2

0

0

2

4

6

8

10

6

8

10

t Fig. 7.4 Case (iii): .M = 1.5

12

U(t)=sin(t−π)+t 10

M=2

8

6

4

2

0

0

2

4

t Fig. 7.5 Case (iv): .M = 2

7.2 The Vickrey Model

281

(ii) .M = 1. .U (τ ) − Mτ has two global minimizers .τ1∗ = π/2 and .τ2∗ = 5π/2, with minimum value .−1. Therefore    U (t) − Mt if 0 ≤ t ≤ π/2 U (τ ) − Mτ = . min 0≤τ ≤t −1 if π/2 < t ≤ 10  q(t) =

.

0

if 0 ≤ t ≤ π/2

sin(t − π ) + 1 if π/2 < t ≤ 10

(iii) .M = 1.5. There exist a local minimum .τ1∗ = 2π/3 with function value ∗ .−1.9132 and a global minimum .τ 2 = 8π/3 with function value .−5.0548. Notice from Fig. 7.4 that a point .p = 5.3876 between .τ1∗ and .τ2∗ yields the same function value as .τ1∗ . Therefore ⎧ ⎪ U (t) − Mt ⎪ ⎪ ⎪ ⎨   −1.9132 . min U (τ ) − Mτ = ⎪ 0≤τ ≤t U (t) − Mt ⎪ ⎪ ⎪ ⎩ −5.0548 ⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎨sin(t − π ) − 0.5t + 1.9132 .q(t) = ⎪ 0 ⎪ ⎪ ⎪ ⎩ sin(t − π ) − 0.5t + 5.0548

if 0 ≤ t ≤ 2π/3 if 2π/3 < t ≤ 5.3876 if 5.3876 < t ≤ 8π/3 if 8π/3 < t ≤ 10 if 0 ≤ t ≤ 2π/3 if 2π/3 < t ≤ 5.3876 if 5.3876 < t ≤ 8π/3 if 8π/3 < t ≤ 10

(iv) .M = 2. Since .U (·) is Lipschitz continuous with constant 2, the minimum of .min0≤τ ≤t {U (τ ) − 2τ } is always attained at the right boundary t. Thus q(t) ≡ 0

.

0 ≤ t ≤ 10

Remark 7.10 The above four solutions for queue sizes satisfy the  ODE everywhere except case 3. From Fig. 7.4 we observe that the quantity . min U (τ ) − Mτ } has a 0≤τ ≤t

kink at point .p = 5.3876. This renders .q(·) not differentiable there. A plot of .q(·) confirms the non-differentiability at .t = 5.3876; see Fig. 7.6.

7.2.5 DAE System Formulation of Dynamic Network Loading The generalized Vickrey model yields a network loading procedure expressible as a system of differential algebraic equations (DAEs). In particular, for every link

282

7 Dynamic Network Loading: Non-physical Queue Models 0.7

Queue Length

0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9

10

t Fig. 7.6 Case (iii): queue size. .q(t) is not differentiable at .t = 5.3876

a ∈ A, the cumulative entering and exiting curves, denoted by .Ua (·) and .Wa (·), are related via the closed-form expression

.

Wa (t) =

.

  mina Ua (τ ) − Ma τ

τ ≤t−t0 ]

where .t0a denotes the constant free flow travel time on link a and .Ma denotes the bottleneck capacity. Given arc .ai ∈ A of every path q = {a1 , a2 , . . . , am(p) }

q∈P

.

in order to incorporate the route-choice information, we associate to the entrance of p p ai the functions .ηai (·) and to the exit of .ai the functions .ξai (·), for all .p ∈ P such that .ai ∈ p. In addition,

.

p

ηai (t) ∈ [0, 1],

.

.

p

ηai (t) ≡ 1,

ai ∈p

p

ξai (t) ∈ [0, 1]

ai ∈ p

p

ξai (t) ≡ 1

ai ∈p

p

In prose, the function .ηai (t) describes for each unit of arc inflow the fraction of p flow associated with path p; similarly, the function .ξai (t) describes for each unit of arc exit flow the fraction of flow associated with path p. Under the first-in-first-out assumption, the following must hold  p p ηai (t) = ξai t + λa1 (t)

.

∀p such that ai ∈ p

(7.106) p

where .λai (t) is the link traversal time when the time of entry is t. Let .uai (t) and p .wai (t) be the arc entering and exiting flows, respectively, associated with path p. As

7.2 The Vickrey Model

283

a consequence of (7.106), we have p

.

uai (t) = q uai (t)

ai ∈q

 p wai t + λai (t)  q wai t + λai (t)

if



q

uai (t) > 0

(7.107)

ai ∈q

ai ∈q

Define  Uai (t) =



t

.

0

Wai (t) =

uai (s) ds,

0

t

wai (s) ds

∀p such that ai ∈ p

The following DAE system for network loading is a direct consequence of the generalized Vickrey model p

p

uai (t) = wai−1 (t) ∀p ∈ P, i ∈ {1, . . . , m(p)}.

p d d Ua (t) = Wa (t) = wa (t) ua (t), ∀a ∈ A. dt dt a∈p   Wa (t) = min a Ua (τ ) − Ma τ ∀a ∈ A.

.

τ ≤t−t0

 1  Ua (t) − Wa (t + t0a ) + t0a ∀a ∈ A. Ma ⎧ 

q p  ⎪ wa t + λa (t) · uau(t) if ua (t) = 0 q ⎪ a (t)  ⎨ a∈q p a∈q

q wa t + λa (t) = ⎪ if ua (t) = 0 ⎪ ⎩0

λa (t) =

(7.108) (7.109) (7.110) (7.111)

∀p ∈ P, a ∈ p

a∈q

(7.112) By convention, p

va0 (t) ≡ hp (t)

.

∀p ∈ P

In the DAE system (7.108)–(7.112), all variables of interest appear explicitly on the left-hand side of the equations. This permits all variables to be solved for sequentially in a pseudo-cascading fashion. Therefore, the existence and uniqueness of solutions to the DAE system are clear. Upon solving the DAE system, one obtains information on .λa (·), the arc traversal time function, for all .a ∈ A. Then the individual path delay may be constructed via . the composition of link exit time functions .τa (t) = t + λa (t) as follows Dp (t, h) = τam(p) ◦ . . . ◦ τa2 ◦ τa1 (t) − t

.

  where .τa2 ◦ τa1 (t) ≡ τa2 τa1 (t) .

∀p ∈ P, p = {a1 , a2 , . . . , am(p) }

284

7 Dynamic Network Loading: Non-physical Queue Models

7.2.6 Continuity of the Effective Path Delay Operator In this section, we will establish continuity of the operator .Ψ : Λ →  2 |P | L+ [t0 , tf ] . These results are essential for the existence proof and algorithm convergence. Notice that the proof provided here works for unbounded path flows and thus is suitable for treating simultaneous route-and-departure-time DUEs, whose demand satisfaction constraints do not imply any upper bounds on the departure rates (see (5.1)–(5.2) in Chap. 5). The next lemma provides a sufficient condition for the continuity of the arc traversal time function .λ(·). Lemma 7.3 Consider an arc with inflow .u(·). Under the generalized Vickrey model expressed in (7.99)–(7.99), the arc traversal time function .λ(·) is continuous if 2 .u(·) ∈ L [t0 , tf ]. Proof Assume that .u(·) ∈ L2 [t0 , tf ]; then .u(·) ∈ L1 [t0 , tf ]. Therefore the cumulative entering vehicle count . U (t) =



t

u(s) ds

.

t0

is absolutely continuous. It is straightforward to verify that the following quantity is continuous.   1 U (t) − M(t) − min {U (τ ) − Mτ } .λ(t) = t0 + τ ≤t M where M denotes the bottleneck capacity and .t0 denotes the constant free flow  time. The next lemma is a technical result that will facilitate the proof of Theorem 7.7. Lemma 7.4 Let .gn (·) : [a1 , b1 ] → [a2 , b2 ], n ≥ 1, be a sequence of functions such that .gn converges to .g(·) : [a1 , b1 ] → [a2 , b2 ] uniformly. In addition, assume 1 .f (·) : [a2 , b2 ] → R is continuous. Then the following uniform convergence holds.     f gn (·) −→ f g(·)

.

n −→ ∞

Proof According to the Heine-Cantor theorem (Royden, 1988), .f (·) is uniformly continuous on .[a2 , b2 ]. It follows that, for every .ε > 0, there exists .δ > 0 such that for any .y1 , y2 ∈ [a2 , b2 ], whenever .|y1 − y2 | < δ, the inequality |f (y1 ) − f (y2 )| ≤ ε

.

7.2 The Vickrey Model

285

holds. Moreover, by the uniform convergence of .gn , there exists some .N > 0 such that, for all .n > N, we have |gn (x) − g(x)| < δ

∀ x ∈ [a1 , b1 ]

.

Thus, for every .n > N,      f gn (x) − f g(x)  ≤ ε

.

∀ x ∈ [a1 , b1 ]



Theorem 7.7 Under the network loading model described in Sect. 7.2.5, the effec |P | , .h → Ψ (·, h) is well-defined tive path delay operator .Ψ : Λ → L2 [t0 , tf ] and continuous. Proof For each .h ∈ Λ, the functions .Ψp (·, h), p ∈ P are uniquely determined by the network loading procedure. To show that the effective path delay operator  |P | is well-defined, it remains to show that .Ψ (·, h) ∈ L2 [t0 , tf ] for each .h ∈ Λ. Notice that there exists an upper bound for the path delays regardless of the network flow profile ⎧

⎨ 1 .Dp (t, h) ≤ ⎩ Ma a∈p

(i,j )∈W

Qij + t0a

⎫ ⎬ ⎭

∀h ∈ Λ, ∀p ∈ P, ∀t ∈ [t0 , tf ] (7.113)

where .Ma and .t0a are the bottleneck capacity and the free flow time, respectively, that are associated with arc a. Recall the definition of the effective path delay

Ψp (t, h) = Dp (t, h) + f t + Dp (t, h) − TA

.

Since .f (·) is continuous, the uniform boundedness of .Dp (t, h), as shown in (7.113), thus implies the uniform boundedness of .Ψp (t, h) for all .h ∈ Λ, p ∈ P and .t ∈  |P | [t0 , tf ]. This leads to the conclusion that .Ψ (·, h) ∈ L2 [t0 , tf ] for all .h ∈ Λ. With the preceding as background, the proof of continuity of the effective delay operator may be given in five parts. Part 1. We first focus on a single link a. For notational convenience, the subscript a will be dropped for now. Consider a sequence of entering flows .uν , ν ≥ 1 that converge to u in the .L2 -norm, that is, . uν − uL2 =



tf

.

t0

1/2 (uν (t) − u(t))2 dt

−→ 0 as ν −→ ∞

286

7 Dynamic Network Loading: Non-physical Queue Models

Consider the cumulative entering vehicle counts ⎧  t ⎪ . ⎪ U (t) = uν (s) ds ⎪ ⎪ ⎨ ν t0 .

ν ≥ 1 t ∈ [t0 , tf ]

 t ⎪ ⎪ . ⎪ ⎪ u(s) ds ⎩U (t) = t0

Then .Uν converges to U uniformly on .[t0 , tf ]; this is due to the following simple observation  t |uν (s) − u(s)| ds ≤ uν − uL1 . |Uν (t) − U (t)| ≤ t0

≤ (t0 − tf )1/2 uν − uL2 −→ 0

(7.114)

where . · L1 is the norm in .L1 [t0 , tf ]. The last inequality of (7.114) is a version of Jenssen’s inequality. . . Part 2. Define .R(τ ) = U (τ ) − Mτ and .Rν (τ ) = Uν (τ ) − Mτ, for τ ∈ [t0 , tf ], where M is the bottleneck capacity. We claim the following uniform convergence .

min {Rν (τ )} −→ min {R(τ )} τ ≤t

τ ≤t

∀ t ∈ [t0 , tf ]

(7.115)

Indeed, for any .ε > 0, by the uniform convergence of .Uν , ν ≥ 1, we can choose N such that for all .ν ≥ N, the following inequality holds .

|Uν (t) − U (t)| ≤ ε

∀ t ∈ [t0 , tf ]

Fix any t, if .ν ≥ N, then .

|Rν (τ ) − R(τ )| = |Uν (τ ) − U (τ )| ≤ ε

∀τ ∈ [t0 , tf ]

(7.116)

Define .τˆ = argminτ ≤t {R(τ )}. By (7.116) we have .

min {Rν (τ )} ≤ Rν (τˆ ) ≤ R(τˆ ) + ε = min {R(τ )} + ε τ ≤t

τ ≤t

(7.117)

On the other hand, define .τˆν = argminτ ≤t {Rν (τ )} for each .ν ≥ 1. Then, given ν ≥ N, it must hold that

.

.

min {R(τ )} ≤ R(τˆν ) ≤ R (ν) (τˆν ) + ε = min {Rν (τ )} + ε τ ≤t

τ ≤t

Taken together, (7.117) and (7.118) imply      . min {Rν } − min {R(τ )} ≤ ε   τ ≤t τ ≤t

∀ν ≥N

(7.118)

7.2 The Vickrey Model

287

Since t is arbitrary, the claim is demonstrated. Part 3. An immediate consequence of (7.99) is the following uniform convergence Wν (t) −→ W (t),

.

qν (t) −→ q(t),

λν (t) −→ λ(t),

τν (t) −→ τ (t) (7.119)

as .ν → ∞, for which we employ notation whose meaning is transparent. The next step is to extend such convergence to the whole network. Consider the sequence of departure rates .hν converging to h in the . · L2 norm. This implies each path departure rate .hp,ν (·) → hp (·) in the . · 2 norm, for all .p ∈ P. A simple induction based on results established in Part 2 yields, as .ν → ∞, the convergences Ua,ν (t) → Ua (t),

.

Wa,ν (t) → Wa (t),

Da,ν (t) → Da (t),

τa,ν (t) → τa (t) (7.120)

uniformly for all .a ∈ A. Part 4. We will show next the uniform convergence of the path delay function .Dp (·, hν ) → Dp (·, h), based on (7.120). Recall the path exit time function τp (t) = τam(p) ◦ . . . ◦ τa2 ◦ τa1 (t)

.

p =



 a1 , a2 , . . . , am(p) ∈ P (7.121)

We start by showing that .τa2 ,ν ◦ τa1 ,ν (t) → τa2 ◦ τa1 (t) uniformly as .ν → ∞. For every .ν ≥ 1, since the inflow of arc .a2 is square-integrable, .τa2 ,ν (·) is continuous by Lemma 7.3. This means that .τa2 (·) is also continuous   since it is the uniform limit of .τa2,ν (·). Lemma 7.4 then implies that .τa2 τa1 ,ν (·) converges uniformly to .τa2 τa1 (·) ; that is, for any .ε > 0, there exists an .N1 > 0 such that, for all .ν > N1 ,      τa τa ,ν (t) − τa τa (t)  < ε/2 2 1 2 1

.

∀ t ∈ [t0 , tf ]

Moreover, there exists some .N2 > 0 such that, for all .ν > N2 ,   τa ,ν (t) − τa (t) < ε/2 2 2

.

∀ t ∈ [t0 , tf ]

Now let .N0 = max{N1 , N2 }. Then, for any .ν > N0 and any .t ∈ [t0 , tf ],      τa ,ν τa ,ν (t) − τa τa (t)  2 1 2 1           ≤ τa2 ,ν τa1 ,ν (t) − τa2 τa1 ,ν (t)  + τa2 τa1 ,ν (t) − τa2 τa1 (t)  .

< ε/2 + ε/2 = ε This shows the desired uniform convergence .τa2 ,ν ◦ τa1 ,ν (t) → τa2 ◦ τa1 (t).

288

7 Dynamic Network Loading: Non-physical Queue Models

The uniform convergence .τp,ν (·) → τp (·) follows immediately by (7.121) and mathematical induction with Lemma 7.4 . As a result, we obtain the uniform convergence of path delay Dp (·, hν ) −→ Dp (·, h)

.

Part 5.

ν → ∞

Finally, recall the definition of the effective delay, namely   Ψ (t, h) = Dp (t, h) + f t + Dp (t, h) − TA

.

Note that .f (·) is continuous; the following uniform convergence follows by Lemma 7.4:     f t + Dp (t, hν ) − TA −→ f t + Dp (t, h) − TA

.

ν → ∞

We conclude that the effective path delay .Ψp (·, hν ) converges uniformly to Ψp (·, h). The desired convergence in the . · L2 norm now follows since the interval .[t0 , tf ] is compact. 

.

In addition to the strong continuity established in the previous theorem, sometimes a different notion of continuity is used, for example, in the proof of solution existence (assumption A3 in Chap. 5). This is recalled below. A3. For any sequence of departure rate vectors .{h(n) }n≥1 ⊂ Λ that are uniformly bounded by a positive constant and converge weakly to .h∗ ∈ Λ, the corresponding effective path delays .Ψp (t, hn ) converge to .Ψp (t, h∗ ) uniformly for all .p ∈ P and .t ∈ [t0 , tf ] as .n → +∞. Theorem 7.8 Under the network loading model described in Sect. 7.2.5, the effec |P | , .h → Ψ (·, h) is continuous in tive path delay operator .Ψ : Λ → L2 [t0 , tf ] the sense of .A3. Proof The proof deviates from that of Theorem 7.7 only in Part 1. In particular, we focus on a single link and consider a uniformly bounded sequence of entering flows .uν , .ν ≥ 1 that converge to u weakly. We define . .Uν (t) =



t

uν (s) ds t0

. U =



t

u(s) ds

∀t ∈ [t0 , tf ]

t0

Then one immediately has that .Uν (·) converges to .U (·) uniformly on .[t0 , tf ]. The rest of the proof is the same as the proof of Theorem 7.7. 

7.3 Some Other Non-physical Queue Models

289

7.3 Some Other Non-physical Queue Models This section briefly covers some other non-physical queue models used in the literature. The first one is an exit flow function model; the second one involves controllable link entrance and exit flows.

7.3.1 Models Based on Link Exit Flow Functions If one posits that it is possible to specify and empirically estimate, or to mathematically derive from some plausible theory, functions that describe the rate at which traffic exits a given network arc for any given volume of traffic present on that arc, one is led to some deceptively simple traffic dynamics. To express this supposition symbolically, we use .xa (t) to denote the volume of traffic on arc a at time t and .ga (xa (t)) to denote the rate at which traffic exits from link a. Where it will not be confusing, we suppress the explicit reference to time t and write the arc volume as .xa and the exit flow function as .ga (xa ) with the understanding that both entities are time varying. It is also necessary to define the rate at which traffic enters arc a, which we denote as .ua (t). Again, when it is not confusing, we may suppress the time dependency of the entrance rate for arc a and simply write .ua . Note that both .ga (xa ) and .ua are rates; that is, they have the units of volume per unit time, so it is appropriate to refer to them as exit flow and entrance flow, respectively. A natural flow balance equation can now be written for each link .

dxa = ua − ga (xa ) dt

∀a ∈ A

(7.122)

where .A denotes the set of all arcs of the network of interest. Although (7.122) is a fairly obvious identity, it seems to have been first studied in depth by Merchant and Nemhauser (1978a) and Merchant and Nemhauser (1978b) in the context of system optimal dynamic traffic assignment. The same dynamics were employed by Friesz et al. (1989) and Wie et al. (1995) to explore certain extensions of the Merchant-Nemhauser model. Exit flow functions have been widely criticized as difficult to specify and measure. Exit flow functions are known to allow certain anomalies as illustrated and discussed by Carey (1986), Carey (1987), Carey (1992) and Carey (1995). As a consequence, many researchers studying dynamic network flow problems have abandoned dynamics based on exit flow functions.

7.3.2 Models with Controlled Entrance and Exit Flows A possible modification of the Merchant-Nemhauser arc dynamics that avoids the use of problematic exit flow functions is to treat both arc entrance and exit flows as

290

7 Dynamic Network Loading: Non-physical Queue Models

control variables. Let .W be the set of origin-destination pairs, and recall .A is the set of arcs for the network of interest. Then, one way to operationalize the idea of modeling entrance and exit flows as controls is to write ij

.

dxa ij ij = ua − va dt

∀a ∈ A, (i, j ) ∈ W

(7.123)

ij

where .xa is the volume on arc a traveling between origin-destination pair .(i, j ), ij ij while .ua and .va denote the rates at which traffic, also traveling between .(i, j ), ij ij enters and exits arc a, respectively. By treating both .ua and .va as control variables, we do not mean to imply that any kind of normative considerations have been introduced, for these variables are viewed as controlled by network users constrained by physical reality and observed only at the level of their aggregate (flow) behavior. A criticism is that missing from the unembellished version of (7.123) is any explanation of the queue discipline for the origin-destination flows on the same arc: just as with dynamics based on exit flow functions, we have no way of ensuring that the FIFO queue discipline is enforced without additional constraints or assumptions. Furthermore, use of dynamics (7.123) without additional constraints may result in flow propagation speeds faster than would occur under free flow with no congestion, a rather profound violation of physical reality. To overcome the difficulties mentioned above, Bernstein et al. (1993), Ran et al. (1996), and Ran and Boyce (1996) have suggested flow propagation constraints for dynamics (7.123) of the form p

p

Ua (t) = Va [t + a (t)]

.

p

p

∀a ∈ A, p ∈ P

(7.124)

where .Ua (.) and .Va (.) are the cumulative numbers of vehicles associated with path p that are entering and leaving link a, respectively, while .a (t) denotes the time needed to traverse link a at time t and .P is the set of all paths. The meaning of these constraints is fairly intuitive: vehicles entering an arc at a given moment in time must exit at a later time consistent with the arc traversal time. Moreover, these constraints assume that flows moving through the network are incompressible; that is, wave packets and vehicle platoons are neither shortened nor elongated in the presence of congestion. It is shown in Friesz et al. (2001) that this incompressibility assumption is incompatible with at least one model of arc delay widely employed in dynamic traffic assignment modeling; this is because the constraints (7.124) omit a fundamental term that describes the expansion and contraction of wave packets or platoons moving through the network.

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Lebacque, J., & Khoshyaran, M. (2002). First order macroscopic traffic flow models for networks in the context of dynamic assignment. In M. Patriksson & K. A. P. M. Labbe (Eds.), Transportation planning state of the art. , Norwell, MA: Kluwer Academic Publishers. LeVeque, R. J. (1992). Numerical methods for conservation laws. Birkhäuser. Li, J., Fujiwara, O., & Kawakami, S. (2000). A reactive dynamic user equilibrium model in network with queues. Transportation Research Part B, 34(8), 605–624. Lighthill, M., & Whitham, G. (1955). On kinematic waves. II. A theory of traffic flow on long crowded roads. Proceedings of the Royal Society of London: Series A, 229, 317–345. Marigo, A., & Piccoli, B. (2008). A fluid dynamic model for T-junctions. SIAM Journal on Mathematical Analysis, 39(6), 2016–2032. Merchant, D., & Nemhauser, G. (1978a). A model and an algorithm for the dynamic traffic assignment problems. Transportation Science, 12(3), 183–199. Merchant, D., & Nemhauser, G. (1978b). Optimality conditions for a dynamic traffic assignment model. Transportation Science, 12(3), 200–207. Moskowitz, K. (1965). Discussion of ‘freeway level of service as influenced by volume and capacity characteristics’ by D.R. Drew and C.J. Keese. Highway Research Record, 99, 43–44. Newell, G. F. (1993a). A simplified theory of kinematic waves in highway traffic. Part I: General theory. Transportation Research Part B, 27(4), 281–287. Newell, G. F. (1993b). A simplified theory of kinematic waves in highway traffic. Part II: Queuing at freeway bottlenecks. Transportation Research Part B, 27(4), 289–303. Newell, G. F. (1993c). A simplified theory of kinematic waves in highway traffic. Part III: Multidestination flows. Transportation Research Part B, 27(4), 305–313. Ni, D., & Leonard, J. (2005). A simplified kinematic wave model at a merge bottleneck. Applied Mathematical Modeling, 29(11), 1054–1072. Nie, X., & Zhang, H. M. (2005). A comparative study of some macroscopic link models used in dynamic traffic assignment. Networks and Spatial Economics, 5(1), 89–115. Pang, J. S., Han, L. S., Ramadurai, G., & Ukkusuri, S. (2011). A continuous-time linear complementarity system for dynamic user equilibria in single bottleneck traffic flows. Mathematical Programming, Series A, 133(1–2), 437–460. Ran, B., & Boyce, D. (1996). Modeling dynamic transportation networks: An intelligent transportation system oriented approach (2nd Ed.). Springer. Ran, B., Boyce, D., & LeBlanc, L. (1996). A new class of instantaneous dynamic user optimal traffic assignment models. Operations Research, 41(1), 192–202. Richards, P. I. (1956). Shockwaves on the highway. Operations Research, 4, 42–51. Royden, H. L., & Fitzpatrick, P. (1988). Real analysis (Vol. 3). Englewood Cliffs, NJ: Prentice Hall. Smoller, J. (1983). Shock waves and reaction-diffusion equations. New York: Springer. Stewart, D. E. (1990). A high accuracy method for solving ODEs with discontinuous right-hand side. Numerische Mathematik, 58, 299–328. Vickrey, W. S. (1969). Congestion theory and transport investment. The American Economic Review, 59(2), 251–261. Walter, W. (1988). Ordinary differential equations. Springer. Wie, B., Tobin, R., Friesz, T., & Bernstein, D. (1995). A discrete-time, nested cost operator approach to the dynamic network user equilibrium problem. Transportation Science, 29(1), 79–92. Wu, J. H., Chen, Y., & Florian, M. (1998). The continuous dynamic network loading problem: A mathematical formulation and solution method. Transportation Research Part B, 32, 173–187. Xu, Y. W., Wu, J. H., Florian, M., Marcotte, P., & Zhu, D. L. (1999). Advances in the continuous dynamic network loading problem. Transportation Science, 33, 341–353. Zhu, D., & Marcotte, P. (2000). On the existence of solutions to the dynamic user equilibrium problem. Transportation Science, 34(4), 402–414.

Chapter 8

Dynamic Network Loading: Physical Queue Models

Another important class of traffic flow models, arguably more realistic than nonphysical queue models, captures the formation, propagation, and dissipation of physical queues and allow spillback to be explicitly modeled. The most widely used physical queue models are the Lighthill-Whitham-Richards model (Lighthill & Whitham, 1955; Richards, 1956), as well as its discrete forms including the cell transmission model (Daganzo, 1994, 1995) and the link transmission model (Yperman et al., 2005; Han et al., 2016).

8.1 The Lighthill-Whitham-Richards Model The Lighthill-Whitham-Richards (LWR) model (Lighthill & Whitham, 1955; Richards, 1956) is a macroscopic link model that describes traffic dynamics in terms of the formation, propagation, and interaction of kinematic waves. It has received increased attention in the field of traffic flow theory in the past several decades due to its capability of capturing key features of vehicular traffic such as shock waves and spillback. The primary mathematical form of the LWR model is a partial differential equation (PDE) describing the temporal-spatial evolution of vehicle density and flow. This PDE is based on the conservation of vehicles (the scalar conservation law) and an explicit density-flow relation known as the fundamental diagram (FD). The reader is referred to Sect. 2.7 for a mathematical review of scalar conservation laws. We consider a homogeneous road segment expressed as a spatial interval .[a, b] ⊂ R. The PDE representation of the LWR model is as follows   ∂t ρ(t, x) + ∂x f ρ(t, x) = 0

.

(t, x) ∈ [0, T ] × [a, b]

© Springer Nature Switzerland AG 2022 T. L. Friesz, K. Han, Dynamic Network User Equilibrium, Complex Networks and Dynamic Systems 5, https://doi.org/10.1007/978-3-031-25564-9_8

(8.1)

295

296

8 Dynamic Network Loading: Physical Queue Models

with appropriate initial and boundary conditions, which will be subsequently discussed in detail when the theory is extended to networks. Here .ρ(t, x) denotes the vehicle density at a given piont in the space-time domain. The fundamental diagram .f (·) : [0, ρj am ] → [0, C] expresses vehicle flow at .(t, x) as a function of .ρ(t, x), where .ρ jam denotes the jam density and C denotes the flow capacity. Throughout this chapter, we impose the following mild assumption on .f (·): (F) The fundamental diagram .f (·) is continuous and concave and vanishes at .ρ = 0 and .ρ = ρ jam . An essential component of the network extension of the LWR model is the specification of boundary conditions at a road junction. Derivation of the boundary conditions should not only take into account physical realism, such as entropy conditions (Garavello & Piccoli, 2006; Holden & Risebro, 1995), but also reflect certain behavioral and operational considerations, such as vehicle turning preferences, driving priorities (Daganzo, 1995; Coclite et al., 2005), and signal controls (Han et al., 2014). Articulation of a junction model is facilitated by the notion of a Riemann Problem, which is an initial-value problem on the junction of interest involving constant initial conditions on each incident link. There exist a number of junction models that yield different solutions of the same Riemann Problem. In one line of research, an entropy condition is defined based on a minimization problem (Holden & Risebro, 1995). In another line of research, the boundary conditions are defined using link demand and supply (Lebacque & Khoshyaran, 1999), which indicate the link’s sending and receiving capacities given the initial conditions. Models following this approach include Daganzo (1995), Jin and Zhang (2003), and Jin (2010). The solution of a Riemann Problem is given by the Riemann Solver (RS), to be elaborated in the next section.

8.1.1 The LWR Model at Road Junctions The network extension of the LWR model will be discussed in this section. As we shall subsequently see, the key to the network extension is the junction model and, in particular, the boundary conditions stipulated for the links incident to the relevant junction. We consider a general road junction J with m incoming roads and n outgoing roads, as shown in Fig. 8.1. At a given road junction, we index its incoming and outgoing links by .{1, . . . , m} and .{m + 1, . . . , m + n}, respectively. In addition, for every .i ∈ {1, . . . , m + n}, the dynamic on link i is governed by the following conservation law   ∂t ρi (t, x) + ∂x fi ρi (t, x) = 0

.

(t, x) ∈ R+ × [ai , bi ]

(8.2)

8.1 The Lighthill-Whitham-Richards Model

m+n

......

m

......

Fig. 8.1 A road junction with m incoming links and n outgoing links

297

J

m+2

2

m+1

1

where link i is expressed as the spatial interval .[ai , bi ], and we will use the subscript i to indicate link-dependent quantities throughout this chapter. The initial condition for this conservation law is ρi (0, x) = ρˆi (x)

.

x ∈ [ai , bi ],

i = 1, . . . , m + n

(8.3)

Notice that the above .(m + n) Cauchy problems (initial value problems) are coupled together via boundary conditions to be specified at the junction. In analogy to the weak solution of the single PDE (8.2), the notion of weak solution for the system of .(m + n) coupling PDEs at the junction is as follows. Definition 8.1 (Weak Solution at a Junction) Let .φi (t, x), i = 1, . . . , m + n be any smooth functions on .R+ × [ai , bi ] with compact support that are also continuously differentiable across the junction, that is φi (t, bi ) = φj (t, aj ),

∂x φi (t, bi ) = ∂x φj (t, aj )

.

(8.4)

for all .i ∈ {1, . . . , m}, j ∈ {m + 1, . . . , m + n}. A weak solution of the system of conservation laws (8.2) and (8.3) is a set of functions .ρi (t, x), i = 1, . . . , m + n, such that m+n   ∞  bi .

0

i=1

 ρi ∂t φi + fi (ρi )∂x φi dxdt +

ai



bi

ρi,0 (x)φi (0, x) dx

= 0

ai

(8.5) for all .φi satisfying (8.4). An immediate consequence of this definition is that any weak solution in the sense of Definition 8.1 satisfies the following flow conservation across junction J : m  .

i=1

  fi ρi (t, bi ) =

m+n  j =m+1

  fj ρj (t, aj )

∀t > 0

(8.6)

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8 Dynamic Network Loading: Physical Queue Models

Equation (8.6) is analogous to the Rankine-Hugoniot condition for scalar conservation laws; we refer the reader to Smoller (1983) for a demonstration. Notice that the Rankine-Hugoniot condition (8.6) amounts to one constraint imposed on the system with .(m + n) variables   fi ρi (t, bi ) i ∈ {1, . . . , m},

  fj ρj (t, bj ) j ∈ {m + 1, . . . , m + n}

.

In other words, this system is largely underdetermined. In order to isolate a unique solution, additional constraints must be introduced. We consider a special type of initial-value problem called the Riemann Problem, which consists of constant initial-value conditions. Definition 8.2 (Riemann Problem at J ) The Riemann Problem at the junction J is defined to be an initial value problem for a network consisting of the single junction J with m incoming links and n outgoing links, all extending to infinity, such that the initial densities are constants on each link:  ρi (0, x) ≡ ρˆi x ∈ (−∞, bi ] i ∈ {1, . . . , m} .

ρj (0, x) ≡ ρˆj

x ∈ [aj , +∞)

j ∈ {m + 1, . . . , m + n}

jam

where .ρˆk ∈ [0, ρk ] are constants for .k = 1, . . . , m + n. The concept of Riemann Solver (RS) is introduced to handle Riemann Problems. A Riemann Solver for the junction J is a mapping that, given any .(m + n)-tuple   of Riemann initial conditions ˆ . ρ , . . . , ρ ˆ , provides a unique .(m + n)-tuple of 1 m+n   boundary conditions . ρ 1 , . . . , ρ m+n such that one can solve the initial-boundary value problem for each link, and the resulting solutions constitute a weak entropy solution of the Riemann Problem at the junction. A precise definition of the Riemann Solver is follows. Definition 8.3 (Riemann Solver) A Riemann Solver for the junction J with m incoming links and n outgoing links is a mapping RS :

m+n

jam

0, ρk

.

k=1



−→

m+n

jam



0, ρk

k=1

    ρˆ1 , . . . , ρˆm+n → ρ 1 , . . . , ρ m+n   that associates with every Riemann initial  data .ρˆ = ρˆ1 , . . . , ρˆm+n a vector of boundary conditions .ρ = ρ 1 , . . . , ρ m+n so that the following holds: (i) The solution of the Riemann Problem restricted on each link k is given by the solution of the initial-boundary value problem with initial condition .ρˆk and boundary condition .ρ k , .k = 1, . . . , m + n.

8.1 The Lighthill-Whitham-Richards Model

299

(ii) The Rankine-Hugoniot condition (flow conservation) holds: m  .

  fi ρ i =

m+n 

  fj ρ j

(8.7)

j =m+1

i=1

(iii) The consistency condition holds:

RS RS[ρ] ˆ = RS[ρ] ˆ

(8.8)

.

Three conditions must be satisfied by the Riemann Solver (RS). Item .(i) above requires that the boundary condition on each link must be properly given so that the initial-value problems not only have well-defined solutions, but these solutions must also be compatible and form a weak entropy solution of the junction. Expression (8.7) simply stipulates the conservation of flow across the junction. Expression (8.8) is a desirable property and is sometimes referred to as the invariance property (Jin, 2010). We will next provide a few examples of Riemann Solvers, followed by a nonmathematical illustration of how they can be used to construction solutions on networks via the wave front tracking procedure (Sect. 8.1.2).

8.1.1.1

Demand and Supply

The articulation of a Riemann Solver is greatly simplified by the notions of demand and supply, introduced by Lebacque and Khoshyaran (1999). For each link i in Fig. 8.1, we let .ρic to be the critical density at which flow is maximized. The demand .Di (t) for each incoming link i and the supply .Sj (t) for each outgoing link j are defined, with respect to the density near the exit and entrance of the link, respectively:  Di (t) = Di (ρi (t, bi −)) =

.

  Sj (t) = Sj ρj (t, aj +) =



Ci   fi ρi (t, bi −) Cj   fj ρj (t, aj +)

if ρi (t, bi −) ≥ ρic if ρi (t, bi −) < ρic

.

(8.9)

if ρj (t, aj +) < ρjc if ρj (t, aj +) ≥ ρjc (8.10)

In prose, the demand represents the maximum flow that can be discharged from the subject link; and the supply represents the maximum flow that can enter the subject link. Notice that both the demand and supply are expressed as functions of density, and they are greater than or equal to the flux function .fi (·) or .fj (·); see Fig. 8.2 for an illustration. In our subsequent presentation, without causing confusion we will use notations .Di (t) and .Di (ρ) interchangeably where the former indicates the

300

8 Dynamic Network Loading: Physical Queue Models

f( ) D( ) S( )

jam

c

0

Fig. 8.2 Demand and supply as functions of density

2

4 6

1

3

5

Fig. 8.3 The diverging (left) and merge (right) junctions

demand as a time-varying quantity and the latter emphasizes demand as a function of density. The same convention will apply to supply. With the notions of demand and supply, a Riemann Solver (RS) may be stated in terms of flow, rather than density, as follows: given the demands of incoming links and supplies of outgoing links at any instance of time, the RS determines the boundary flows at all the links. We next present two Riemann Solvers (RS) for the diverge and merge junctions shown in Fig. 8.3, respectively. These RSs are proposed and analyzed independently in Daganzo (1995) and Garavello and Piccoli (2006).

8.1.1.2

Riemann Solver for the Diverge Junction

We first consider the diverge junction in Fig. 8.3, with the single incoming link 1 and two outgoing links 2 and 3. The demand .D1 (t) and the supplies, .S2 (t) and .S3 (t),

8.1 The Lighthill-Whitham-Richards Model

301

are defined by (8.9) and (8.10), respectively.1 The Riemann Solver for this junction relies on the following two conditions: (C1) Vehicles leaving link 1 headed to 2 and 3 according to some fixed turning percentage. (C2) Subject to (C1), the flow through the junction is maximized. The diverge junction model, described by (C1) and (C2), may be explicitly written as   S2 S3 f1out = min D1 , , α1,2 α1,3 (8.11) . in f2 = α1,2 · f1out f3in = α1,3 · f1out where .f1out denotes the exit flow of link 1 and .fjin denotes the entering flow on link j = 2, 3. The complete Riemann Solver is constructed as follows. Given the Riemann initial conditions .ρˆ1 , ρˆ2 , ρˆ3 , we compute the corresponding demand .D1 of link 1 and supplies .S2 , S3 of links .2, 3. We then determine the boundary flows .f1out , f2in , f3in according to (8.11). Then, the vehicle densities adjacent to the junction are given as

.

ρ1

.

⎧   out ⎪ ⎪ ⎨max ρ : fi (ρ) = f1 = ρˆ1 ⎪ ⎪ ⎩ρ c 1

ρi =

.

⎧   out ⎪ ⎪ ⎨min ρ : fi (ρ) = fi ρic ⎪ ⎪ ⎩ρˆ i

if f1out < D1 if f1out = D1 and ρˆ1 < ρ1c if f1out = D1 and ρˆ1 ≥ ρ1c if fiout < Si if fiout = Si and ρˆi < ρic if fiout = Si and ρˆi ≥ ρic

for .i = 2, 3. This defines the mapping .(ρˆ1 , ρˆ2 , ρˆ3 ) → (ρ 1 , ρ 2 , ρ 3 ), which gives rise to the Riemann Solver for the diverge node.

8.1.1.3

Riemann Solver for the Merge Junction

We now turn to the merge junction in Fig. 8.3, with two incoming links 4 and 5 and one outgoing link 6. In view of this merge junction, assumption (C1) becomes irrelevant as there is only one outgoing link; and assumption (C2) cannot determine

1 For simplicity of notations, we will sometimes drop the explicit dependence of the demand and supply on the density.

302

8 Dynamic Network Loading: Physical Queue Models

(a)

(b)

Fig. 8.4 Illustration of the merge model. The shaded areas represent the feasible region expressed by (8.12). (a): Rule (R1) is compatible with (C2); and there exists a unique point Q satisfying both (C2) and (R1). (b): (R1) is incompatible with (C2); in this case, the model selects point Q within the set .Ω, which is closest to the line through the origin with slope . 1−p p

a unique solution.2 To address this issue, we consider the parameter .p ∈ (0, 1) and the following driving priority rule: (R1) The actual link exit flows satisfy .(1 − p) · f4out (t) = p · f5out (t). However, (R1) may be incompatible with assumption (C2); see Fig. 8.4 for an illustration. Whenever there is a conflict between (R1) and (C2), we will respect (C2) and relax (R1) so that the solution is chosen to be the point that is closest to the line .y = 1−p p x among all the points yielding the maximum flow. Clearly, such a point is unique. Mathematically, we let .Ω be the set of points .(f4out , f5out ) that solves the following maximization problem  .

max f4out + f5out such that f4out ∈ [0, D4 (t)], f5out ∈ [0, D5 (t)], f4out + f5out ≤ S6 (t) (8.12)

 .  Moreover, we define the ray .R = (f4out , f5out ) ∈ R2+ : (1 − p) · f4out = p · f5out . Then the solution of the merge model is defined to be the projection of R onto .Ω; that is, (f4out,∗ , f5out,∗ ) =

.

2 More

argmin (f4out , f5out )∈Ω



d (f4out , f5out ), R

(8.13)

generally, as pointed out by Coclite et al. (2005), when the number of incoming links exceeds the number of outgoing links, (C1) and (C2) combined are not sufficient to find a unique solution.

8.1 The Lighthill-Whitham-Richards Model

303



where .d (f4out , f5out ), R denotes the Euclidean distance between .(f4out , f5out ] and the set R

d (f4out , f5out ), R =

.

  min (f4out , f5out ) − (x, y)2

(x, y)∈R

One can similarly construct the complete Riemann Solver by following the procedure at the end of Sect. 8.1.1.2.

8.1.2 The LWR Model on Road Networks The Riemann Problems and Riemann Solvers serve as building blocks of a complete LWR solution on a road network. To construct such a solution, a final ingredient is the so-called wave-front tracking (WFT) method. The WFT method was originally proposed by Dafermos (1972) as an approximation scheme for the following Cauchy problem

.

   ∂t ρ(t, x) + ∂x f ρ(t, x) = 0 ρ(0, x) = ρ(x) ˆ

(8.14)

where the initial condition .ρ(·) ˆ is assumed to have bounded variation (BV) (Bressan, 2000). WFT approximates the initial condition .ρ(·) ˆ using piecewise constant (PWC) functions and approximates .f (·) using piecewise affine (PWA) functions. It is an event-based algorithm which resolves a series of wave interactions, each expressed as a Riemann Problem (RP). The WFT method is primarily used for showing existence of weak solutions of conservation laws by successive refinement of the initial data and the flux function; see Bressan (2000) and Holden and Risebro (2002). Garavello and Piccoli (2006) extend the WFT to treat the network case and show the existence of the weak solution on a network. We provide a brief description of this procedure below. Fix a Riemann Solver (RS) for each road junction in the network. Consider a family of piecewise constant approximations .ρˆiε (x) of the initial condition .ρˆi (x) on each link and a family of piecewise affine approximations of the fundamental diagrams .fiε (ρ), where .ε is a parameter such that .ρˆiε (x) → ρˆi (x) and .fiε (ρ) → fi (ρ) as .ε → 0. A WFT approximate solution on the network is constructed as follows: 1. Within each link, solve a Riemann Problem at each discontinuity of the PWC initial data. At each junction, solve a Riemann Problem with the given RS. 2. Construct the solution by gluing together the solutions of individual RPs up to the first time when two traveling waves interact or when a wave interacts with a junction.

304

8 Dynamic Network Loading: Physical Queue Models

3. For each interaction, solve a new RP, and prolong the solution up to the next time of any interaction. 4. Repeat processes 2–3. To show that the procedure described above indeed produces a well-defined approximate solution on the network, one needs to ensure that the following three quantities are bounded: (1) the total number of waves; (2) the total number of interactions (including wave-wave and wave-junction interactions); and (3) the total variation (TV) of the piecewise constant solution at any point in time. These quantities are known to be bounded in the single conservation law case; in fact, they all decrease in time (Bressan, 2000). However, for the network case, one needs to proceed carefully in estimating these quantities as they may increase as a result of a wave interacting with a junction, which may produce new waves in all other links incident to the same junction. The reader is referred to Garavello and Piccoli (2006) for a more detailed discussion of these interactions. For a sequence of approximate WFT solutions .ρ ε , .ε > 0. If one can show that the total variation of .ρ ε is uniformly bounded, then, as .ε → 0, a weak entropy solution on the network is obtained.

8.1.3 Partial Differential Algebraic Equation System Formulation of Dynamic Network Loading The aim of this section is to formulate the LWR-based dynamic network loading (DNL) problem as a system of partial differential algebraic equations (PDAEs). The proposed PDAE system uses vehicle density and queues as the primary unknown variables and computes link dynamics, flow propagation, and path delay for any given set of departure rates along utilized paths. The PDAE system captures vehicle spillback explicitly and accommodates a wide range of junction types and Riemann Solvers. We fix a network .G(A, V) expressed as a directed graph with .A being the set of links and .V being the set of nodes. Let .P be the set of paths employed by travelers and .W be the set of origin-destination pairs. Each path .p ∈ P is expressed as an ordered set of links p = {1, 2, . . . , m(p)}

.

where .m(p) is the number of links in this path. There are several crucial components of a complete network loading procedure, each of which will be elaborated in a subsection below.

8.1 The Lighthill-Whitham-Richards Model

8.1.3.1

305

Within-Link Dynamics

For each .i ∈ A, the link dynamic is governed by the scalar conservation law   ∂t ρi (t, x) + ∂x ρi (t, x) · vi ρi (t, x) = 0

.

(t, x) ∈ [0, T ] × [ai , bi ] (8.15)

subject to initial conditions and boundary conditions to be determined in the next  subsection. The fundamental diagram .fi (ρi = ρi · vi ρi satisfies condition (F) stated at the beginning of Sect. 8.1. In order to explicitly incorporate drivers’ route p choices, for every .p ∈ P such that .i ∈ p, we introduce the function .μi (t, x), .(t, x) ∈ [0, T ] × [ai , bi ], which represents, in every unit of flow .fi (ρi (t, x)), the fraction associated with path p. We will subsequently call these variables path disaggregation variables (PDV). For each moving car, its surrounding traffic has a certain composition, distinguished by path (e.g., 20% following path .p1 , 30% following path .p2 , 50% following path .p3 ). As this car moves, such a composition will not change since its surrounding traffic all move at the same speed under the first-in-first-out (FIFO) principle (i.e., no overtaking). In mathematical terms, p this means that the path disaggregation variables, .μi (·, ·), are constants along the trajectories of cars .(t, x(t)) in the space-time diagram, where .x(·) is the trajectory of a moving car on link i. That is, .

 d p μi t, x(t) = 0 dt

∀p such that i ∈ p,

which, according to the chain rule, becomes  d p p ∂t μi t, x(t) + ∂x μi (t, x) · x(t) = 0, dt

.

which further leads to another set of partial differential equations on link i   p p .∂t μ (t, x) + vi ρi (t, x) · ∂x μ (t, x) = 0 ∀p such that i ∈ p i i

(8.16)

where .ρi (t, x) is the solution of (8.15). The following obvious identity holds  p . μi (t, x) = 1 whenever ρi (t, x) > 0 (8.17) p i

where .p i means “path p contains (or traverses) link i” and the summation appearing in (8.17) is with respect to p. By convention, if .ρi (t, x) = 0, then p .μ (t, x) = 0 for all p. i 8.1.3.2

Determination of Boundary Conditions at an Ordinary Node

A crucial component of the network extension of the LWR model is the specification of boundary conditions for all links incident to a given node or junction. This

306

8 Dynamic Network Loading: Physical Queue Models

problem may be further complicated by the possibility that the node/junction is also the origin or destination of some trips. This issue will be addressed by introducing virtual links, as elaborated later in Sect. 8.1.3.3. Here in this section, we assume the case where the node of interest is neither the origin nor the destination of any paths. Such a node is coined ordinary node, and we use .V o to represent the set of ordinary nodes in the network. For a given ordinary node J , the conservation laws on all incident links are coupled together through a given junction model, i.e., a Riemann Solver. We denote by .I J and .OJ the set of incoming and outgoing links of node J , respectively. A common prerequisite for applying the Riemann Solver for the ordinary junction J is the determination of the flow distribution (turning percentage) matrix (Garavello p & Piccoli, 2006), which relies on knowledge of .μi (t, bi ) for all .i ∈ I J . We define the time-dependent matrix associated with J AJ (t) =



.

 J J αijJ (t) ∈ [0, 1]|I |×|O |

(8.18)

where, by convention, we use subscript i for incoming links and j for outgoing links. Each element .αijJ (t) of the matrix .AJ (t) represents the turning percentages of cars discharged from i that enter downstream link j . For all p that traverses J , the following holds 

αijJ (t) =

p

∀ i ∈ I J , j ∈ OJ

μi (t, bi )

.

(8.19)

p i, j

 It can be easily verified that .αijJ (t) ∈ [0, 1] and . j αijJ (t) ≡ 1 according to (8.17). We are now ready to express the boundary conditions for the ordinary junction o .J ∈ V . Let RS

.

AJ

|I J |+|O J |

:

[0,

jam ρk ]

k=1

→

|I J |+|O J |

jam

[0, ρk ]

k=1 jam

be a given Riemann Solver, where .ρk denotes the jam density of link k. Notice that the dependence of the Riemann Solver on .AJ has been indicated with a superscript. The boundary conditions for PDEs (8.15) read J

ρk (t+, bk ) = RSkA

.

AJ

ρl (t+, al ) = RSl

     ρi (t, bi ) i∈I J , ρj (t, aj ) j ∈OJ

     ρi (t, bi ) i∈I J , ρaj (t, aj ) j ∈OJ

∀k ∈ I J . (8.20) ∀l ∈ OJ (8.21)

J

where .RSkA [·] denotes the k-th component of the mapping, .k = 1, . . . , |I J | + |OJ |.

8.1 The Lighthill-Whitham-Richards Model

307

Remark 8.1 mean that, given the current traffic states   (8.20)–(8.21)  Intuitively,  ρi (t, bi ) i∈I J and . ρj (t, aj ) j ∈OJ immediately adjacent to the junction J at

.

J

time t, the Riemann Solver .RS A specifies, for each incident link k or l, the corresponding boundary conditions .ρk (t+, bk ) or .ρl (t+, al ) for time .t+. In prose, at each time instance, the Riemann Solver inspects the traffic conditions near the junction and proposes discharging (receiving) flows of its incoming (outgoing) links for the next time instance. And this is done in a way consistent with the vehicle turning percentages given by .AJ . Most Riemann Solvers also incorporate some traffic control or prioritization strategies. Furthermore, since the Riemann Solver operates with the full knowledge of every link incident to the junction, the downstream (upstream) boundary condition of an incoming (outgoing) link is determined jointly by all the other links connected to the same junction. This underlying mechanism effectively couples a number of LWR PDEs together, which is highly challenging from the points of view of analysis and computation. Accordlingly, the upstream boundary conditions (8.16) associated with PDEs are determined as  p  fi ρi (t, bi ) · μi (t, bi ) p   .μ (t, aj ) = ∀p such that {i, j } ⊂ p, ∀j ∈ OJ j fj ρj (t, aj ) (8.22)   p where the numerator .fi ρi (t, bi ) · μi (t, bi ) expresses the exit flow on link i associated with path p, which, by flow conservation, is equal to the entering flow of link j associated with the same path p; the denominator represents the total entering flow of link j . Remark 8.2 Unlike the density-based PDE, the .μ-based PDE does not have any downstream boundary condition due to the fact that the path-disaggregation variables have only non-negative travel speeds—same as car speeds—and do not spillover. They can be interpreted as Lagrangian labels that travel with the fluid particles (cars).

8.1.3.3

Determination of Flow Distribution at Origin/Destination Nodes

The case where a road junction is also the origin/destination of some paths is complicated since the flow conservation constraint (8.7) no longer holds: vehicles either are “generated” (if it is an origin) or “vanish” (if it is a destination). A simple and effective way to circumvent this issue is to introduce virtual links. A virtual link is an imaginary road segment between a non-ordinary node and an origin/destination; see Fig. 8.5. By adding virtual links to original network, we ˜ V) ˜ in which all road junctions are ordinary obtain an augmented network .G(A, and, hence, fall within the scope of the previous section.

308

8 Dynamic Network Loading: Physical Queue Models

s

t J

J

Fig. 8.5 Illustration of the virtual links. Left: a virtual link connecting an origin (s) to an ordinary junction J . Right: a virtual link connecting a destination (t) to an ordinary junction J

˜ V). ˜ For Let us denote by .S the set of origins in the augmented network .G(A, o any .o ∈ S, we denote by .P ⊂ P the set of paths that originate from s and by .Io the virtual link incident to this origin. For each .p ∈ P o we denote by .hp (t) the departure rate (path flow) along p. It is expected that a buffer (point) queue may form at o in case the receiving capacity of the downstream .Io is insufficient to accommodate all  the departure rates . p∈P o hp (t). For this buffer queue, denoted .qo (t), we employ a Vickrey-type dynamic (Vickrey, 1969); that is, ⎧ ⎪ ⎨So (t)  if qo (t) > 0 d  qo (t) = . hp (t) − min hp (t), So (t) if qo (t) = 0 ⎪ dt ⎩ p∈P o o 

(8.23)

p∈P

where .So (t) denotes the supply of the virtual link .Io . The only difference between (8.23) and Vickrey’s model is the time-varying downstream receiving capacity provided by the virtual link. It remains to determine the dynamics for the path disaggregation variables p (PDV). More precisely, we need to determine .μo (t, ao ) for the virtual link .Io where o .p ∈ P , and .x = ao is the upstream boundary of .Io . This will be achieved using the Vickrey-type dynamic (8.23) and the FIFO principle. In particular, we define the queue exit time function .λo (t) where t denotes the time at which drivers depart and join the point queue, if any; .λo (t) expresses the time at which the same group of drivers exit the queue. Clearly, FIFO dictates that 

t



.

 hp (τ ) dτ =

0 p∈P o

λo (t)

  fo ρo (τ, ao ) dτ

o∈S

(8.24)

0

where the two integrands on the left- and right-hand sides of the equation are flow into the queue and flow leaving the queue, respectively. We may determine the path disaggregation variables as  hp (t) p μo λo (t), ao =  p∈P o hp (t)

.

(8.25)

8.1 The Lighthill-Whitham-Richards Model

309

 Note that, if . p∈P o hp (t) = 0, then the flow leaving the point queue at time .λo (t) is also zero; thus there is no need to determine the path disaggregation variables. Therefore, the identity (8.25) is well-defined and meaningful.

8.1.3.4

Calculation of Path Travel Times

In light of the preceding discussion, we may express the path travel times which are the output of a full DNL model. The path travel time consists of link travel times plus possible queuing time at the origin. We define the link exit time function .λi (t) for any .i ∈ A using the following expression:  .

t

  fi ρi (t, ai ) dt =

0



λi (t)

  fi ρi (t, bi ) dt

(8.26)

0

For a path expressed as .p = {1, 2, . . . , m(p)}, the travel time along this path is calculated as λo ◦ λ1 ◦ λ2 . . . ◦ λm(p) (t)

.

(8.27)

. where .f ◦ g(t) = g(f (t)) denotes the composition of two functions.

8.1.3.5

The PDAE System

We are now ready to present a generic PDAE system for the dynamic network loading procedure. Let us begin with some key notation. G(A, V) VL ˜ V) ˜ .G(A, .S .P o .P 0 .V J .I J .O J .A AJ .RS . .

the original network with link set .A and node set .V the set of virtual links the augmented network including virtual links the set of origins the set of all paths in the network the set of paths originating from .o ∈ S the set of ordinary junctions in the augmented network the set of incoming links of a junction .J ∈ V the set of outgoing links of a junction .J ∈ V the flow distribution matrix associated with junction J the Riemann Solver for junction J , which depends on .AJ

We also list some key variables of the PDAE system below. hp (t) ρi (t, x) p .μ (t, x) i . .

the path departure rate along .p ∈ P the vehicle density on link .i ∈ A the proportion of flow on link i associated with path p

310

8 Dynamic Network Loading: Physical Queue Models

the point queue at the origin .o ∈ S the point-queue exit time function at origin .o ∈ S  Given any vector of path departure rates .h = hp (·) : p ∈ P), the proposed PDAE system then reads qo (t) λo (t)

. .

⎧ ⎪ ⎨So (t)  qo (t) > 0 dqo (t)  = hp (t) − min . hp (t), So (t) qo (t) = 0 ⎪ dt ⎩ p∈P o o 

∀o ∈ S.

p∈P

(8.28) 

t





hp (τ ) dτ =

0 p∈P o



t

λo (t)

  fo ρo (τ, ao ) dτ

  fi ρi (t, ai ) dt =



0

λi (t)

  fi ρi (t, bi ) dt

(8.29)

∀i ∈ A˜.

(8.30)

0

  ∂t ρi (t, x) + ∂x ρi (t, x) · vi ρi (t, x) = 0 p ∂t μi t,

∀o ∈ S.

0





x + vi ρi (t, x)



p · ∂x μi t,



x =0

(t, x) ∈ [0, T ] × [ai , bi ]. (8.31) (t, x) ∈ [0, T ] × [ai , bi ]. (8.32)

hp (t) ∀o ∈ S. p∈P o hp (t)   p fi ρi (t, bi ) μi (t, bi ) p   μj (t, aj ) = ∀p ⊃ {i, j }. fj ρj (t, aj )  p   AJ (t) = αijJ (t) , αijJ (t) = μi (t, bi ) ∀i ∈ I J , j ∈ OJ .

p μo (λo (t), ao ) = 

(8.33)

(8.34) (8.35)

p i, j

J

ρk (t+, bk ) = RSkA

J

ρl (t+, al ) = RSlA

     ρi (t, bi ) i∈I J , ρj (t, aj ) j ∈OJ

     ρi (t, bi ) i∈I J , ρj (t, aj ) j ∈OJ

Dp (t, h) = λo ◦ λ1 ◦ λ2 . . . ◦ λm(p) (t)

∀p ∈ P,

∀k ∈ I J . (8.36) ∀l ∈ OJ .

∀t ∈ [0, T ]

(8.37) (8.38)

Equation (8.28) describes the (potential) queuing process at each origin. Equations (8.29) and (8.30) express the queue exit time function for the point queue at the origin s and the link exit time function for a link, respectively. Equations (8.31)– (8.32) express the link dynamics in terms of car density .ρ and the PDV; Eqs. (8.33)– (8.38) specify the upstream boundary conditions for the PDV, as these variables can only propagate forward in space. Equations (8.35)–(8.37) determine the boundary conditions at junctions. Finally, Eq. (8.38) determines the path travel times.

8.2 Variational Formulation of the LWR Model

311

The above PDAE system involves the partial differential operators .∂t and .∂x . Solving such a system requires solution techniques from the theory of numerical partial differential equations (PDE) such as finite difference methods (Godunov, 1959; LeVeque, 1992) and finite element methods (Larson & Thomée, 2005).

8.2 Variational Formulation of the LWR Model The partial differential equation of the LWR type (8.2) admits a variational solution representation as extensively discussed in the mathematical literature (Le Floch, 1988; Lax, 1973; Aubin et al., 2008; Evans, 2010). The most widely recognized form in the traffic modeling community is known as the generalized Lax-Hopf formula (Newell, 1993a; Claudel & Bayen, 2010a,b; Han et al., 2016), which is elaborated in Sect. 2.8. The variational formulation of the LWR model enables us to derive a differential algebraic equation (DAE) system formulation of the network loading procedure. Compared to the PDAE system presented in Sect. 8.1.3, the DAE system has the following distinct features: (1) it eliminates the partial differential operator; (2) the primary variable is flow instead of density; (3) it introduces simplified boundary conditions at junctions and eliminates virtual links; and (4) it facilitates efficient numerical computations while preserving all the modeling features of the PDAE system.

8.2.1 Notation and Brief Recap of Variational Theory We start with the following notions concerning the variational principle. Let .N(t, x) be the cumulative number of vehicles that have passed location x on a road segment by time t. It is straightforward to verify that   ∂t N(t, x) = f ρ(t, x) ,

.

∂x N(t, x) = −ρ(t, x)

This leads to the following Hamilton-Jacobi equation   ∂t N(t, x) − f − ∂x N(t, x) = 0

.

t ∈ [0, T ], x ∈ [0, L]

Let .f in (t) and .f out (t) be the inflow and outflow of the road, respectively. We further define the cumulative (upstream) entering and (downstream) exiting vehicle counts  N up (t) =

t

.

t0

 f in (s) ds

N dn (t) =

t

t0

f out (s) ds

312

8 Dynamic Network Loading: Physical Queue Models

We also define the initial condition for the Hamilton-Jacobi equation, .N ini (x). As a DNL model typically assumes empty network at initial time .t0 , we have that N ini (x) ≡ 0

.

∀x ∈ [0, L]

for all links in the network. With such a simplification, we reduce the generalized Lax-Hopf formula (2.184)–(2.187) to the following N(t, x) = tf ∗ (v) = 0

.

N(t, x) =

(t, x) ∈ ΩI .

min B(u; t, x)

u∈[ xt , v]

N (t, x) = tf ∗ (v) = 0  N (t, x) = min

(8.39)

(t, x) ∈ ΩI I .

(8.40)

(t, x) ∈ ΩI I I .

min B(u; t, x),

u∈[ xt , v]



min u∈[−w,

x−L t ]

C(u; t, x)

(8.41) (t, x) ∈ ΩI V (8.42)

where  x x ∗ + f (u). B(u; t, x) = N up t − u u   x − L x−L ∗ dn t− + f (u) C(u; t, x) = N u u

.

(8.43) (8.44)

v = f  (0+), w = f  (ρ jam −), and .f ∗ (·) is the concave transformation of the fundamental diagram .f (·)

.

f ∗ (u) =

.

sup

ρ∈[0, ρj ]



f (ρ) − uρ



8.2.2 Dynamics at the Origin Nodes In contrast to the PDAE system, here we no longer distinguish ordinary and nonordinary nodes; hence virtual links are no longer needed for modeling purposes. This is due to the fact that the variational formulation treats flow (or quantities of the same nature such as demand and supply) as the primary variable. For a given origin node .o ∈ S, we let the point queue be .qo (t) and let .j ∈ Oo be an outgoing link of node o. The Vickrey type dynamic reads .

 d qo (t) = hp (t) − foout (t) dt o p∈P

(8.45)

8.2 Variational Formulation of the LWR Model

313

where .P o denotes the set of paths originating from o. Due to the interaction of the point queue with links incident to the origin node, we employ the following Riemann Solver

     out Ao Di (t) i∈I o , Do (t), Sj (t) j ∈Oo .fo (t) = RSo where the demand of the point queue is defined as ⎧ ⎪ qo (t) > 0 ⎨M  .Do (t) = hp (t) qo (t) = 0 ⎪ ⎩ p∈P o

where .M is a sufficiently large number. Moreover, elements of the matrix .Ao related to the origin are given as 

p∈P o ,p j



αoj =

.

p∈P o

hp (t)

j ∈ Oo

hp (t)

Next, we determine the queuing delay at the origin. Following the first-in-firstout principle, we have 

t





λo (t)

hp (s) ds =

.

0 p∈P o

0

foout (s) ds

up

By introducing the convenient notation of .No (t) and .Nodn (t), we can re-write the queuing delay as up

No (t) = Nodn (λo (t))

.

8.2.3 Differential Algebraic Equation System Formulation with General Fundamental Diagram The first DAE system we shall present assumes a quite general form of the fundamental diagrams for links; namely, they are continuously differentiable and strictly concave and vanish at zero and jam densities. The complete DNL procedure consists of two main subproblems.

314

8 Dynamic Network Loading: Physical Queue Models

8.2.3.1

Determination of Link Demand and Supply

In a LWR-type DNL model, to realistically capture inter-link congestion propagation (e.g., spillback), it is essential to identify the demand and supply (Lebacque & Khoshyaran, 1999) of links adjacent to a particular junction, so that a specific junction model can be applied to determine the inflows/outflows of relevant links. The junction model is also frequently referred to as the Riemann Solver in the mathematical literature (Garavello et al., 2016). This section aims to formulate the link demand and supply using the Lax-Hopf solution representation. The simplified Lax-Hopf formulae (8.39)–(8.42) lead to the following proposition: Proposition 8.1 Consider a dynamic network loading procedure with time horizon [0, T ]. Given any link, for any .t ∈ [0, T ], the demand .D(t) and the supply .S(t) of this link can be expressed as

.

 S(t) =

.

D(t) =

.

if N up (t) < K r (t)    −1 f (f ) (ur ) if N up (t) = K r (t)

C

 C



f (f  )−1 (ul )



if N dn (t) < K l (t) if N dn (t) = K l (t)

(8.46)

(8.47)

where     L dn ∗ N t+ − Lf (u)/u . .K (t) = min u u∈[−w,− Lt ]     L + Lf ∗ (u)/u K l (t) = min N up t − u u∈[ Lt , v] r

(8.48)

(8.49)

and .ur = argminK r (t), .ul = argminK l (t). Here we stipulate the following u

u

generalization of the function .(f  )−1 , in case .f (·) has a linear piece or is not continuously differentiable: .∀u ∈ [f  (0+), f  (ρ jam −)], (f  )−1 (u) = ρ ∗

.

where ρ ∗ satisfies f  (ρ ∗ +) < u ≤ f  (ρ ∗ −)

Proof Since the demand or supply profile is determined completely by the location of the separating shock wave (Bretti et al., 2006),3 it suffices for us to keep track

3 A separating shock wave is a generalized characteristic that separates the whole link into one uncongested region and one congested region. Notice that spillback occurs when the separating shock reaches the entrance of the link.

8.2 Variational Formulation of the LWR Model

315

of conditions that indicate whether or not the separating shock wave reaches both boundaries of the link by means of the Lax-Hopf formula (8.39)–(8.42). In particular, we notice from (8.42) that the separating shock reaches the entrance if and only if min B(u; , t, 0) ≥

.

u∈[0, v]

min u∈[−w,

−L t ]

C(u; , t, 0)

or

N up (t) ≥ K r (t)

or

N dn (t) ≥ K l (t)

and the separating shock reaches the exit if and only if min B(u; , t, L) ≤

.

u∈[ Lt , v]

min

u∈[−w, 0]

C(u; , t, L)

When the first (or second)  case happens,    a unique characteristic  line with slope .ur L L (or .ul ) connects .(t, 0) to . t − ur , L . or (t, L) to t − ul , 0 . Notice that such a characteristic line can be a normal one or part of a rarefaction wave. Thus, it follows from standard conservation law theory that      L = f (f  )−1 (ur ) . .S(t) = f (ρ(t, 0+)) = f ρ t− ,L ur      L = f (f  )−1 (ul ) D(t) = f (ρ(t, L−)) = f ρ t − , 0) ul

(8.50) (8.51)

This proves the following: S(t) =

.

 C

if N up (t) < K r (t)   −1  f (f ) (ur ) if N up (t) ≥ K r (t)

 D(t) =

.

(8.52)

if N dn (t) < K l (t)   −1  f (f ) (ul ) if N dn (t) ≥ K l (t)

C

(8.53)

Finally, we observe that  N (t) = N (t, 0) = min

.

up

 min B(u; t, 0),

u∈[0, v]

min u∈[−w,





= min N (t), up

min

u∈[−w, − Lt ]

N

dn

−L t ]

C(u; t, 0)

L t+ u



L − f ∗ (u) u



  = min N up (t), K r (t) ⇒ N up (t) ≤ K r (t) Similarly, we also have .N dn (t) ≤ K l (t). This completes the proof.



316

8 Dynamic Network Loading: Physical Queue Models

8.2.3.2

Determination of Boundary Conditions Using Demand and Supply

The following notation is recalled: up

Ni (t) :

the cumulative entering vehicle count on link i ∈ A

Nidn (t) :

the cumulative exiting vehicle count on link i ∈ A

.

fiin (t) : fiout (t)

the flow into link i ∈ A

:

the flow out of link i ∈ A

We consider an arbitrary junction, which may be incident to virtual links. The definition of a Riemann Solver for such junction often requires knowledge of the distribution matrix .AJ (t) as defined in (8.18). In order to determine .αijJ (t) for any incoming link .i ∈ I J and outgoing link .j ∈ OJ , we invoke the following observation p

p

μi (λi (t), Li ) = μi (t, 0)

.

(8.54)

where .λi (t) denotes the link exit time function given the entry time t; see (8.26). Equation (8.54) is consistent with conservation of vehicles and the first-in-first-out (FIFO) principle. For the link exit time function, we define its inverse function .τi (t), which represents the entry time of a car that exits link i at time t. In order to determine the function .τi (t), we invoke the common technique of up measuring the horizontal difference between the two curves .Ni (t) and .Nidn (t) up

Nidn (t) = Ni (τi (t))

.

∀i ∈ A

(8.55)

Therefore, the identity (8.54) can be rewritten as p

p

μi (t, Li ) = μi (τi (t), 0)

.

∀i ∈ p

(8.56)

Consequently, the entry .αijJ (t) of the distribution matrix .AJ (t) reads αijJ (t) =



.

p

μi (τi (t), 0)

∀i ∈ I J , j OJ

(8.57)

p i,j

In order to apply a specific junction model, we employ the Riemann Solver (RS) that takes into account the demands and supplies of adjacent links and determines the inflows and outflows of the corresponding links (see, e.g., Sects. 8.1.1.2 and 8.1.1.3). Similar to (8.20) and (8.21), we use the Riemann Solvers to determine J

fkin (t+) = RSkA

.

     Di (t) i∈I J , Sj (t) j ∈OJ

∀k ∈ OJ .

(8.58)

8.2 Variational Formulation of the LWR Model J

flout (t+) = RSlA

317

     Di (t) i∈I J , Sj (t) j ∈OJ

∀l ∈ I J

(8.59)

where the superscript .AJ is used to emphasize the dependence of the RS on the distribution matrix.

8.2.3.3

The DAE System

With the same notation introduced in Sect. 8.1.3.5, the DAE system for the DNL problem can be summarized as follows: .

 d qo (t) = hp (t) − foout (t) ∀o ∈ S. dt o

(8.60)

p∈P

 Cj Sj (t) =   fj (fj )−1 (ur )  Ci Di (t) =   fi (fi )−1 (ul ) up

up

if Nj (t) < Kjr (t) up

if Nj (t) = Kjr (t) if Nidn (t) < Kil (t) if Nidn (t) = Kil (t)

(8.62)

.

up

Nidn (t) = Ni (τi (t)) ,

(8.61)

.

Ni (t) = Nidn (λi (t))

∀i ∈ A ∪ S.

(8.63)

∀p such that {i, j } ⊂ p.

(8.64)

p

(8.65)

p

p

μj (t, 0) =

fiout (t)μi (τi (t), 0)

  AJ (t) = αijJ (t) ,

fjin (t) αijJ (t) =



μi (τi (t), 0) .

p i,j

     Di (t) i∈I J , Sj (t) j ∈OJ ∀k ∈ OJ . fkin (t) = RSk

    J  Di (t) i∈I J , Sj (t) j ∈OJ ∀l ∈ I J . flout (t) = RSlA

    o  ∀o ∈ S. foout (t) = RSoA Di (t) i∈I o , Do (t), Sj (t) j ∈Oo AJ

d up N (t) = fiin (t), dt i

d dn N (t) = fiout (t) dt i

Dp (t, h) = λo ◦ λ1 ◦ λ2 . . . λm(p) (t) − t

∀i ∈ A ∪ S.

p = {1, 2 . . . , m(p)}

(8.66) (8.67) (8.68) (8.69) (8.70)

Equations (8.60)–(8.70) form the DAE system for the dynamic network loading procedure. The key input variables are the path departure rates .h = (hp (t) : p ∈ P, t ∈ [0, T ]), and the main outputs are the path travel times .(Dp (t, h) : p ∈ P, t ∈ [0, T ]). Compared to the PDAE system, the DAE system involves only time derivatives and no spatial derivatives. The whole system may be discretized in time and allows calculation in a time-forward fashion.

318

8 Dynamic Network Loading: Physical Queue Models

The DAE system also involves a function-inversion problem arising from (8.63), as we defined link traversal times via the horizontal difference between the two cumulative curves. However, .τi (t) and .λi (t) may be easily calculated numerically up in a discrete-time setting or by linear interpolation of .Ni (t) and .Nidn (t) in a continuous-time setting. The DAE system requires solving minimization problems involved in the evaluation of .Kjr (t) and .Kil (t), which are given by (8.48) and (8.49), respectively. In general, these minimization problems are continuous if the fundamental diagram is piecewise smooth. In the following section, we consider a special case where the fundamental diagram is piecewise affine.

8.2.4 Differential Algebraic Equation System Formulation with Triangular Fundamental Diagram The DAE system can be further simplified provided that the fundamental diagram is triangular. In this case, the minimization problems (8.48) and (8.49) reduce to a discrete form, as shown in the below proposition.

8.2.4.1

Determination of Link Demand and Supply

Proposition 8.2 Assume that the fundamental diagram is triangular; that is,  f (ρ) =

.

ρ ∈ [0, ρc )

vρ −w(ρ

− ρ jam )

ρ ∈ [ρ c , ρ jam ]

Then the expressions for supply (8.61) and demand (8.62) become (here the link indices i, j are omitted) 

  L if N up (t) < N dn t − w + ρ jam L . .S(t) =     L L f out t − w if N up (t) = N dn t − w + ρ jam L    C if N up t − Lv > N dn (t) D(t) =     f in t − Lv if N up t − Lv = N dn (t) C

(8.71)

(8.72)

where C, ρ c , ρ jam , L denote link flow capacity, critical density, jam density, and length, respectively. Proof In the case of the triangular fundamental diagram, its concave transformation becomes f ∗ (u) = C − ρc u u ∈ [−w, v]

.

8.2 Variational Formulation of the LWR Model

319

According to (8.48), we have K r (t) =

.

    L LC N dn t + − + Lρ c u u u∈[−w, − Lt ] min

(8.73)

Differentiating the quantity above to be minimized w.r.t. u yields f

.

out

        L LC L L L out t+ t+ · − 2 + 2 = 2 · −f + C ≥ 0, u u u u u

for almost every t. This implies that the minimum in (8.73) is obtained at u = ur = −w, which leads to     LC L L r + + ρ jam L .K (t) = Ndn t − + Lρ c = Ndn t − w w w Meanwhile, when N up (t) = K r (t), the link entrance is in the congested phase. Kinematic  wave theory implies that the supply is equal to f (ρ(t, a+)) = L f out t − w . This proves (8.71). The proof of (8.72) is entirely similar and omitted here.  In the triangular case, the decision variable u (from the definitions of K r (t) and K l (t)), which essentially represents admissible wave speeds, can only take two values; also see Sect. 2.8.2.2 for more insights. In this case, the proposed DAE system becomes a continuous-time version of the link transmission model (Yperman et al., 2005); a proof will be presented later in Sect. 8.4.3.

8.2.4.2

The DAE System

With the same notation introduced in Sect. 8.1.3.5, the DAE system for the DNL problem can be summarized as follows: .

 d qo (t) = hp (t) − foout (t) ∀o ∈ S. dt o

(8.74)

p∈P

⎧ ⎨Cj

  L up jam if Nj (t) < Njdn t − wjj + ρj Lj     Sj (t) = L up jam ⎩f out t − Lj if Nj (t) = Njdn t − wjj + ρj Lj j wj ⎧   up ⎨ Ci if Ni t − Lvii > Nidn (t)     . Di (t) = up Li dn ⎩f in t − Li if N t − i i vi vi = Ni (t) up

Nidn (t) = Ni (τi (t)) ,

up

Ni (t) = Nidn (λi (t))

∀i ∈ A ∪ S.

.

(8.75)

(8.76) (8.77)

320

8 Dynamic Network Loading: Physical Queue Models p

p

μj (t, 0) =

fiout (t)μi (τi (t), 0)

  AJ (t) = αijJ (t) ,

fjin (t) αijJ (t) =



∀p such that {i, j } ⊂ p.

(8.78)

p

(8.79)

μi (τi (t), 0) .

p i,j

     Di (t) i∈I J , Sj (t) j ∈OJ ∀k ∈ OJ .

    J  Di (t) i∈I J , Sj (t) j ∈OJ ∀l ∈ I J . flout (t) = RSlA

    o  ∀o ∈ S. foout (t) = RSoA Di (t) i∈I o , Do (t), Sj (t) j ∈Oo J

fkin (t) = RSkA

d up N (t) = fiin (t), dt i

d dn N (t) = fiout (t) dt i

Dp (t, h) = λo ◦ λ1 ◦ λ2 . . . λm(p) (t) − t

∀i ∈ A ∪ S.

p = {1, 2 . . . , m(p)}

(8.80) (8.81) (8.82) (8.83) (8.84)

The DAE system (8.74)–(8.84) can be discretized and solved in a time-forward fashion. It allows efficient computation since the variational/minimization formulation of the LWR model is reduced to a minimal form, as conveyed by the “if” statements in (8.75)–(8.76). A numerical procedure for solving this DAE system was implemented in Matlab with open-source codes available at https://github.com/ DrKeHan/DTA.

8.3 The Cell Transmission Model Daganzo (1994) and Daganzo (1995) introduce the cell transmission model (CTM) as a discrete-time approximation of the kinematic wave model. The CTM is by far the most widely used traffic flow model in the literature due to its simple representation of the rather complex propagation and interaction of kinematic waves and its straightforward network extensions that capture shock formation and vehicle spillback. The model is based on a spatial partition of a link into small intervals called cells and the assumption that the fundamental diagram is trapezoidal.

8.3.1 Link Dynamic of the CTM The CTM is based on a spatial partition of the road segment of interest into cells and on a trapezoidal fundamental diagram illustrated in Fig. 8.6.

8.3 The Cell Transmission Model Fig. 8.6 The trapezoidal fundamental diagram employed by the cell transmission model

321

f( ) Q j (t)

v

-w

0

jam

To illustrate this model in detail, we introduce the following notation: j: t: .nj (t): .yj (t):

index for the cell index for the discrete time the number of vehicles in cell j during time t the number of vehicles entering cell j from its upstream neighbor cell .j − 1 during time t the maximum number of vehicles that can be discharged into cell j from cell .j − 1 during time t the holding capacity of cell j (in number of vehicles) the speed of forward kinematic waves (see Fig. 8.6) the speed of backward kinematic waves (see Fig. 8.6)

Qj (t):

.

Nj (t): v: w:

.

The fundamental recursion that propagates traffic through the cells reads nj (t + 1) = nj (t) + yj (t) − yj +1 (t).   w Nj (t) − nj (t) yj (t) = min nj −1 (t), Qj (t), v

.

(8.85) (8.86)

Equation (8.85) simply stipulates that the change in the cell occupancy during t is equal to the difference of the incoming flow and the outgoing flow. Equation (8.86) poses several restrictions on the flow that can enter cell j from cell .j − 1 at time t; namely, it must be the minimum of three quantities: the number of cars in cell .j − 1 at time t, the maximum flow that can be discharged from cell .j − 1, and   w . v Nj (t) − nj (t) , which takes into account the blockage effect of cell j when it is congested. The recursive relationship then facilitates the propagation of trafficrelated quantities in the space-time diagram of the road segment of interest, subject to some initial-boundary conditions specified a priori.

322

8 Dynamic Network Loading: Physical Queue Models

8.3.2 Network Extension of the CTM The CTM allows extensions to traffic networks through straightforward bookkeeping. Specifically, the network under consideration is represented as a directed graph consisting of nodes and links. Each link is associated with a trapezoidal fundamental diagram with given physical data. Moreover, each link is partitioned uniformly into sub-intervals or cells. The most important entity of the network extension is the junction model. In the original work on the CTM (Daganzo, 1995), only simple merge and diverge junction topologies, shown in Fig. 8.7, were considered. In general, junctions involving more incoming and outgoing links can be reduced to a combination of these two basic junction types. Moreover, other junction models (Riemann Solvers) can be easily incorporated into the CTM framework; thus the simple merge and diverge junctions are not essential to our discussion here.

8.3.2.1

Ordinary Links

By definition, an ordinary link connects two cells that belong to the same road. If we let link k connect cells .j − 1 and j and .yk (t) be the flow on link k as time t

a c b

(Merge junction)

(Cell representation)

e d f

(Diverge junction)

(Cell representation)

Fig. 8.7 The merge and diverge junctions considered by the CTM

8.3 The Cell Transmission Model

323

advances to .t + 1, then the following identity, equivalent to (8.86), is obtained    wj  Nj (t) − nj (t) yk (t) = min nj −1 (t), Qj −1 (t), Qj (t), vj

.

(8.87)

where .wj and .vj are associated with cell j . Equation (8.87) can be re-written as yk (t) = min{Dj −1 (t), Sj (t)}

.

(8.88)

where the demand .Dj −1 (t) of the upstream cell and the supply .Sj (t) of the downstream cell are defined as   Dj −1 (t) = min Qj −1 (t), nj −1 (t) .    wj  Nj (t) − nj (t) . Sj (t) = min Qj (t), vj

.

(8.89) (8.90)

Dj −1 (t) represents the maximum flow that can be sent from cell .j − 1, and .Sj (t) represents the maximum flow that can be received by cell j ; both quantities depend on time t obviously.

.

8.3.2.2

Merge Junction

We consider the merge junction shown in Fig. 8.7 and its cell representation. We let ya→c (t) and .yb→c (t) be the flows between relevant cells, where the meaning of the employed notation is obvious. Then the demand and supply of cells give rise to the following constraints

.

ya→c (t) ≤ Da (t),

.

yb→c ≤ Db (t)

(8.91)

and ya→c (t) + yb→c (t) ≤ Sc (t).

.

(8.92)

Note that alone the constraints above are insufficient to isolate a unique solution for the junction model. To achieve this, three additional considerations are imposed: R1. The flow through the merge junction is maximized subject to (8.91) and (8.92). R2. There exist driving priority parameters .pa > 0 and .pb > 0 such that .pa +pb = 1, and for every unit of flow entering the downstream cell c, .pa of it is sent from cell a and .pb of it is sent from cell b. R3. In the event that R1 is conflicting with R2, respect R1 and relax R2. More precisely, the solution will be the point that approximates R1 best among all the points that maximize the flow.

324

8 Dynamic Network Loading: Physical Queue Models

(a)

(b)

Fig. 8.8 Diagrams describing the merge junction model for the CTM

These three additional rules can be explained with the help of Fig. 8.8. The shaded area indicates the feasible region once constraints (8.91) and (8.92) are applied. In addition, we let the set .Ω contain those points that maximize the flow through the junction subject to R1 and show it as a thick line segment in the figure. In the scenario shown in Fig. 8.8a, there exists a unique point .Q ∈ Ω that also satisfies R2; and the coordinates of Q are chosen as the solution of this merge junction model. In the scenario depicted in Fig. 8.8b, however, no point can simultaneously satisfy R1 and R2. In this case, we invoke R3 and find the point .Q ∈ Ω that is the closest to the line .yb→c = ppab ya→c . Notice that such a point is unique as it is the minimum-norm projection of the line .yb→c = ppab ya→c onto the convex set .Ω. Consequently, we let .Q be the solution. The merge junction model discussed here for the CTM is a discrete version of the Riemann Solver for continuous-time LWR model presented in Sect. 8.1.1.3.

8.3.2.3

Diverge Junction

We turn to the diverge junction depicted in Fig. 8.7 and discuss the rule of discharging cars from the upstream cell d into downstream cells e and f . We assume that cars released from cell d enter e and f according to an a priori turning ratio .αd,e > 0 and .αd,f > 0, which satisfy .αd,e + αd,f = 1. The solution is obtained by solving the following maximization problem .

subject to

max yd→e (t) + yd→f (t). ⎧ ⎨ yd→e (t) + yd→f (t) ≤ Dd (t) (t) ≤ Se (t), yd→f (t) ≤ Sf (t) y ⎩ d→e αd,f · yd→e (t) = αd,f (t) · yd→e (t)

(8.93) (8.94)

8.4 The Link Transmission Model

325

that admits a closed-form solution, as follows   Se (t) .yd→e (t) = αd,e min Dd (t), . αd,e   Sf (t) yd→f (t) = αd,f min Dd (t), . αd,f

(8.95) (8.96)

It can be seen that the diverge junction model discussed here for the CTM is a discrete version of the Riemann Solver for continuous-time LWR model presented in Sect. 8.1.1.2.

8.4 The Link Transmission Model Yperman et al. (2005) propose the discrete-time link transmission model (LTM) based on Newell’s famous trilogy (Newell, 1993a,b,c). In analogy to the cell transmission model, the LTM treats an entire link as a “cell" and determines its sending and receiving capacities using cumulative vehicle counts and the variational theory. It relies on the triangular fundamental diagram shown in Fig. 8.9 for the propagation of link flow.

8.4.1 Link Dynamics of the LTM Let us define, for each link, the cumulative entering vehicle count .N up (t) and the cumulative exiting vehicle count .N dn (t), where t is a discrete-time index with step Fig. 8.9 The triangular fundamental diagram employed by the LTM

f( ) C

k

0

-w

jam

326

8 Dynamic Network Loading: Physical Queue Models

size denoted by .δ. We define the maximum number of vehicles that can be sent by this link during time interval .[t, t + δ] to be   L S boundary (t) = N up t + δ − − N dn (t) k

.

(8.97)

where k denotes the free flow speed of the link, which coincides with the forward kinematic wave speed (see Fig. 8.9). Notice that .S boundary (t) represents vehicle volume, not flow, because of the discrete-time nature of the model. The maximum number of vehicles that can leave this link during period .[t, t + δ] is given as S link (t) = C · δ

.

(8.98)

where C denotes the link’s flow capacity. As a result, the number of vehicles sent during .[t, t + δt] is the minimum of the two   S(t) = min S boundary (t), S link (t)

.

(8.99)

Similarly, on the receiving side of the model, we have   L R boundary (t) = N dn t + δ − + ρ jam L − N up (t) w

.

R link (t) = C · δ   R(t) = min R boundary (t), R link (t)

(8.100)

where .ρ jam and L denote the jam density and length, respectively; w denotes the speed of backward kinematic waves. The above quantities .S(t) and .R(t) represent the sending and receiving capacities of the link, respectively, and are analogous to the notions of demand and supply (8.89)-(8.90) defined for the cells.

8.4.2 Network Extension of the LTM Having developed the notions of demand and supply for the link, the extension of the LTM to a network is relatively straightforward and follows closely the models described above for the CTM. However, one should note that the LTM is represented primarily by vehicle counts, that is, by .N up (t) and .N dn (t), which should be updated when the discrete time advances. As an example, we consider a sequence of two links, 1 and 2, shown in Fig. 8.10. up Let .Ni (t) and .Nidn (t) be the vehicle counts associated with .i = 1, 2. The actual number of vehicles that advance from link 1 into link 2 during .[t, t + δ], denoted by

8.4 The Link Transmission Model

327

1

Fig. 8.10 A sequence of two links

2

y1→2 (t), is given as the minimum of the sending capacity of link 1 and the receiving capacity of link 2. That is,

.

y1→2 (t) = min {S1 (t), R2 (t)}

.

(8.101)

where .S1 (t) denotes the sending capacity (demand) of 1 at time t; .R2 (t) denotes the receiving capacity (supply) of 2 at time t. Accordingly, we update the vehicle count functions as follows: N1dn (t + δ) = N1dn (t) + y1→2 (t).

.

up

up

N2 (t + δ) = N2 (t) + y1→2 (t)

(8.102) (8.103)

For merge and diverge junctions, the determination of flows sent and received by relevant links is similar to the procedure detailed in Sect. 8.3.2. One needs to invoke the notions of demand and supply from (8.99) and (8.100) to apply such junction models. This procedure is relatively straightforward and will not be elaborated here.

8.4.3 Relationship with the Continuous-Time Variational Formulation Having been derived from Newell’s variational principles (Newell, 1993a,b,c), the LTM is closely related to the variational formulation presented in Sect. 2.8.2.2. This section will verify that under an appropriate time discretization scheme, the continuous-time variational formulation (2.189)–(2.191) with a triangular fundamental diagram reduces to the LTM. Recall that in the link transmission model, the sending capacity .S(t) given by (8.99) and the receiving capacity .R(t) given by (8.100) of any link both represent traffic volume in a single discrete time interval. Dividing .S(t) by the time step .δ (which should approach zero when converging to a continuous-time version) and using the forward discretization scheme  f

.

in

L t− v



    N up t + δ − Lv − N up t − Lv ≈ , δ

328

8 Dynamic Network Loading: Physical Queue Models

we get     N up t + δ − Lv − N dn (t) S(t) . = min ,C δ δ      N up t + δ − Lv − N up t − Lv = min δ    N up t − Lv − N dn (t) + ,C δ       up t − L − N dn (t) N L v = min f in t − + , C v δ      f in t − Lv if N up t − Lv = N dn (t) =   C if N up t − Lv > N dn (t). Similarly, dividing .R(t) by the time step .δ and using the forward discretization scheme       L L − N dn t − w N dn t + δ − w L out .f ≈ , t− w δ we derive that   N dn t + δ − R(t) . = min δ   N dn t + δ − = min



+ ρ jam L − N up (t) ,C δ    L dn t − L w −N w δ    L N dn t − w + ρ jam L − N up (t) + ,C δ

 =

 f out t −

C

L w



L w

 if N dn t −  if N dn t −



L jam L w +ρ  L jam L w +ρ



= N up (t) > N up (t).

which are recognized as the continuous-time formulations of the demand and supply functions shown in Corollary (8.2). Therefore, the LTM can be regarded as a special case of the time-discretized DAE system (8.60)–(8.69) based on a forward timediscretization scheme.

8.5 Continuity of the Effective Delay Operator for LWR-Based Dynamic. . .

329

8.5 Continuity of the Effective Delay Operator for LWR-Based Dynamic Network Loading This section presents a rigorous continuity result for the path delay operator based on the LWR network model that explicitly captures physical queues and vehicle spillback. In showing the desired continuity, we propose a systematic and general approach for analyzing the well-posedness of two specific junction models: the diverge and the merge models presented in Sects. 8.1.1.2 and 8.1.1.3, respectively. The underpinning analytical framework employs the wave-front tracking methodology (Dafermos, 1972; Holden & Risebro, 2002) and the technique of generalized tangent vectors (Bressan, 1993; Bressan et al., 2000). A major portion of our proof involves the analysis of the interactions between kinematic waves and junctions. Such analysis is further complicated by the fact that vehicle turning percentages at a diverge node are determined endogenously by drivers’ route choices within the DNL procedure. As a result, special tools are developed to handle this unique situation. As we shall later see, a crucial step of the above process is to estimate and bound from below the minimum network supply, which is defined in terms of local vehicle densities in the same way as in Fig. 8.2. In fact, if the supply somewhere tends to zero, the well-posedness of the diverge junction may fail, as we demonstrate in Sect. 8.5.1. This has also been confirmed by the earlier study of Szeto (2003), where a wave of jam density is triggered by a red signal light and causes spillback at the upstream junction, leading to a jump in path travel times. Remarkably, we are able to show that (1) if the supply is bounded away from zero, then the diverge junction model is well-posed and (2) the desired boundedness of the supply is a natural consequence of the dynamic network loading procedure that involves only the simple merge and diverge nodal models. This is a highly non-trivial result because it not only plays a role in the continuity proof but also implies that gridlock can never occur in the network loading procedure in a finite time horizon.

8.5.1 Well-Posedness of Junction Models In the mathematical modeling of a physical system, the term well-posedness refers to the property of having a unique solution, and the behavior of that solution hardly changes when there is a slight change in the initial/boundary conditions. The wellposedness of the Cauchy problem (initial value problem) for scalar conservation laws has been shown in Bressan (2000); a variety of well-posedness problems for different junction models have been analyzed in Garavello and Piccoli (2009). In the context of traffic network modeling, well-posedness is a desirable property of network performance models capable of supporting analyses and computations of DTA models. It is also closely related to the continuity of the path delay operator. This section investigates the well-posedness (i.e., continuous dependence on the initial/boundary conditions) of two specific junction models. These two junctions

330

8 Dynamic Network Loading: Physical Queue Models

2

4

1

6

5

3 Fig. 8.11 The diverging (left) and merge (right) junctions

are depicted in Fig. 8.11, and their corresponding diverge and merge rules (Riemann Solvers) are described in Sects. 8.1.1.2 and 8.1.1.3, respectively. We begin with the diverge junction model. Unlike previous studies on the well-posedness of junction models (Garavello et al., 2016; Han et al., 2016), a major challenge is to incorporate drivers’ route choices, expressed by the path disaggregation variable .μ, into the model and our analysis. In effect, we need to establish the continuous dependence of the system on the initial/boundary conditions for both .ρ and .μ. As we shall see in Sect. 8.5.1.1, such continuity does not hold in general. Following this, Sect. 8.5.1.3 provides a sufficient condition for continuity to hold. And this sufficient condition is the key for showing the desired continuity of the delay operator.

8.5.1.1

An Example of Ill-posedness

It has been shown by Han et al. (2016) that well-posedness holds if the vehicle turning percentages, .α1,2 and .α1,3 , are time-independent and nonzero. However, such an assumption does not hold for DNL models as the turning percentages vary and are endogenous to the PDAE system; see (8.35). As a result well-posedness does not occur, which is demonstrated by the following example. For simplicity, assume the same fundamental diagram .f (·) and thus the same .ρ c , jam .ρ and C for all links 1, 2, and 3. We consider a series of Riemann initial data parameterized by .ε on the three links ρ1 (0, x) = ρˆ1 ∈ (ρ c , ρ jam ) . ρ2 (0, x) = ρ ˆ2ε ∈ (ρ c , ρ jam ) ρ3 (0, x) = ρˆ3ε ∈ (0, ρ c )

(8.104)

∀x ∈ [0, Li ], such that

.

    f ρˆ2ε = εf ρˆ1 ,

.

    f ρˆ3ε = (1 − ε)f ρˆ1

(8.105)

8.5 Continuity of the Effective Delay Operator for LWR-Based Dynamic. . .

331

Fig. 8.12 An example of ill-posedness of the Cauchy problem with established route choices. Left: junction topology. Right: constant initial densities on each link

where .ε ≥ 0 is a parameter. Such a data configuration implies that link 1 and link 2 are both in the congested phase, while link 3 is in the uncongested phase; see Fig. 8.12. Two paths exist in this example: .p1 = {1, 2} and .p2 = {1, 3}. Let us also specify p p the initial conditions for .μ1 1 and .μ1 2 , which arise from travelers’ route choices: p

μ1 1 (0, x) = ε

x ∈ [0, L1 ]

p μ1 2 (0,

x ∈ [0, L1 ]

.

x) = 1 − ε

Notice that the above initial data are also considered part of the initial conditions of the Cauchy problem. We will next analyze solutions of such a family of Cauchy problems, as .ε tends to zero. • When .ε > 0, we claim that the initial conditions .ρˆ1 , ρˆ2ε , and .ρˆ3ε satisfying (8.104)–(8.105) constitute a constant weak solution at the junction. To see this, we follow the junction model (8.11) and the definitions of demand and supply (8.9)–(8.10) to get out .f1 (t)

    f (ρˆ2ε ) C S2 (t) S3 (t) = min C, , , = min D1 (t), α12 α13 ε 1−ε   C = f (ρˆ1 ). = min C, f (ρˆ1 ), (8.106) 1−ε

f2in (t) = εf1out (t) = εf (ρˆ1 ) = f (ρˆ2ε ).

(8.107)

f3in (t) = (1 − ε)f1out (t) = (1 − ε)f (ρˆ1 ) = f (ρˆ3ε )

(8.108)

where C denotes the flow capacity. Thus .ρˆ1 , .ρˆ2ε , and .ρˆ3ε form a constant weak solution at the junction.

332

8 Dynamic Network Loading: Physical Queue Models

• When .ε = 0, the turning rates satisfy .α12 (t) ≡ 0, .α13 (t) ≡ 1. Effectively link 1 is only connected to link 3. We easily deduce that f1out (t) = C.

(8.109)

f2in (t) = 0.

(8.110)

f3in (t) = C

(8.111)

.

As a result, on link 1, a backward-propagating rarefaction wave with left and right states .ρˆ1 and .ρ c , respectively, is created. On link 3, a forward-propagating rarefaction wave with left and right states .ρ c and .ρˆ30 is created. Link 2 remains in a completely jammed state with full density .ρ jam . It is interesting to compare (8.106)–(8.108) with (8.109)–(8.111), which are derived from two scenarios with an infinitesimal difference. This comparison reveals the jumps in .f1out (t) and .f3in (t), from .f (ρˆ1 ) or .f (ρˆ3ε ) to C, as .ε tends to zero. This is a clear indication of the discontinuous dependence of the diverge junction model on its initial conditions. Let us take a closer look at the mechanism that triggers such discontinuity. According to (8.106), the expression for .f1out (t) when .ε > 0 is     f (ρˆ2ε ) C εf (ρˆ1 ) , = min C, f1out (t) = min C, ε 1−ε ε

.

As long as .ε is positive, the fraction . εf (ερˆ1 ) ≡ f (ρˆ1 ) < C. However, when .ε = 0 we have an expression of “. 00 ,” which should be equal to .∞ since link 2 has effectively no influence on the junction, and we have .min{C, 00 } = C. Therefore, .f1out (t) jumps from .f (ρˆ1 ) to C as .ε becomes 0. Remark 8.3 The fact that .ρˆ2ε tends to .ρ jam as .ε → 0 plays a key role in this example. As we shall later see, bounding .ρˆ2ε away from the jam density would guarantee the continuous dependence.

8.5.1.2

Generalized Tangent Vectors

To prepare for the well-posedness result in the next section, we provide a brief introduction to the technique of generalized tangent vectors here, while referring the reader to Garavello and Piccoli (2006) for a more elaborate discussion. The technique of generalized tangent vectors was proposed by Bressan (1993) for showing the well-posedness of conservation laws and used later by Garavello and Piccoli (2006) to show the well-posedness of junction models in connection with the LWR model. Its description follows.

8.5 Continuity of the Effective Delay Operator for LWR-Based Dynamic. . .

333

Given a piecewise constant function .F (x) : [a, b] → R, a tangent vector is defined in terms of the shifts of the discontinuities of .f (·). More precisely, let us indicate by .{xk }N k=1 the discontinuities of F where .a = x0 < x1 < · · · < xN < xN +1 = b and by .{Fk }N k=1 the values of F on .(xk−1 , xk ). A tangent vector of .F (·) is a vector .ξ = (ξ1 , . . . , ξN ) ∈ RN such that for each . > 0, one may define the corresponding perturbation of .F (·), denoted by .F (·) and given by F (x) = Fk

x ∈ [xk−1 + ξk−1 , xk + ξk )

.

for .k = 1, . . . , N + 1, where we set .ξ0 = ξN +1 = 0. The norm of the tangent vector ξ is defined as

.

N .  ξ  = |ξk | · |F (xk +) − F (xk −)|

.

(8.112)

k=1

In other words, the norm of the tangent vector is the sum of each .|ξi | multiplied by the magnitude of the shifted jumps. The process of showing the well-posedness of a junction model using the tangent vectors is as follows. Given piecewise constant initial/boundary conditions on each link incident to that junction, one considers their tangent vectors. By showing that the norms of their tangent vectors are uniformly bounded in time among approximate wave-front tracking solutions, it is guaranteed that the .L1 distance of any two solutions is bounded, up to a constant, by the .L1 distance of their respective initial/boundary conditions. More precisely, we have the following theorem: Theorem 8.1 If the norm of a tangent vector does not increase in time among all approximate wave-front tracking solutions, then the junction model is well-posed and has a unique solution. 

Proof See Garavello and Piccoli (2006).

Throughout our analysis, we use the notation .(ρi , ρi− ) to denote a wave interacting with the junction from road i, where .ρi− is the density value in front of the wave (in the same direction as the traveling wave) and .ρi is the density behind the wave. After the interaction, a new wave may be created on some road j , and we use .ρj− and .ρj+ to denote the density at J on road j before and after the interaction, respectively. Lemma 8.1 Consider the diverge junction. If a wave .(ρi , ρi− ) on i interacts with J , then the shift .ξj produced on j , as a result of the shift .ξi on i, satisfies    Δqj  − ξj ρj+ − ρj− = ξi ρi − ρi+ Δqi

.

(8.113)

334

8 Dynamic Network Loading: Physical Queue Models

where Δqi = fi (ρi− ) − fi (ρi+ ),

.

Δqj = fj (ρj+ ) − fj (ρj− )

the flow jumps in these two waves. Proof To fix the idea, assume that i is the incoming road 1 (the other cases can be treated in a similar way). Let .(ρ1 , ρ1− ) be the interacting wave where .ρ1− is in front of the wave in the same direction as the traveling wave. The solution at the junction after the interaction is defined by computing out,+ .q 1

  S2 S3 , = min D1 (ρ1 ), α1,2 α1,3

where .ρ1+ is such that .f (ρ1+ ) = q1out,+ . If .q1out,+ = f (ρ1 ), then on road 1 a backward wave .(ρ1 , ρ1+ ) is generated (where .ρ1 is in front of the wave) with either out,+ .f (ρ1 ) < q ≤ q1out,− or .f (ρ1 ) > q1out,+ ≥ q1out,− . In other words, part of the 1 change in the flow caused by the interaction passes through the junction, and part of it is reflected back onto link 1. We can then conclude the lemma following the same estimate as in Garavello and Piccoli (2006).  Definition 8.4 If a .ρ-wave from i interacts with the junction, producing a .ρ-wave on link j , we call i and j the source and recipient of this interaction, respectively. Such an event is denoted .i → j . We observe that .|ξi (ρi− −ρi )| is precisely the .L1 -distance of two initial conditions on i with and without the shift .ξi at the discontinuity .(ρi , ρi− ), respectively. Similarly, .|ξj (ρj+ − ρj− )| is the .L1 -distance of the two solutions on j as a result of having or not having the initial shift .ξi on i, respectively. Furthermore, if .ξi is the only shift in the initial condition on i, then .|ξi (ρi− − ρi )| is the norm of the tangent vector for i (see definition (8.112)) before the interaction, and .|ξj (ρj+ − ρj− )| is the norm of the tangent vector for j after the interaction. From (8.113), we have .

   |Δqj |   −   + ξi ρi − ρi  ξj ρj − ρj−  = |Δqi |

(8.114)

Therefore, to bound the norm of the tangent vectors, one has to check that the |Δq | multiplication factors . |Δqji | remain uniformly bounded, regardless of the number of interactions that may occur at this junction. Let us now consider the diverge junction. Assume a wave interacts with J from road i and produces a wave on road j , where .i, j = 1, 2, 3. According to (8.11), .Δq2 = α1,2 Δq1 and .Δq3 = α1,3 Δq1 . Consequently, we have the following matrix

8.5 Continuity of the Effective Delay Operator for LWR-Based Dynamic. . .

335

|Δqi | of multiplication factors . |Δq : j|

|Δq1 | = 1, |Δq1 | .

|Δq2 | |Δq3 | = α1,2 , = α1,3 ; |Δq1 | |Δq1 |

|Δq1 | 1 |Δq2 | , = = 1, |Δq2 | α1,2 |Δq2 |

|Δq3 | α1,3 ; = |Δq2 | α1,2

(8.115)

|Δq1 | 1 |Δq2 | α1,2 |Δq3 | = = = 1. , , |Δq3 | α1,3 |Δq3 | α1,3 |Δq3 | We denote this matrix by .{Qij }. According to Garavello and Piccoli (2006), in order to estimate the tangent vector norm, it suffices to keep track of just one single shift and show the corresponding tangent vector norm is bounded regardless of the number of wave interactions. To this end, the only meaningful sequence of wave interaction is of the form .i → j , .j → k, . . . . We observe that, if .α1,2 and .α1,3 are constants, .Qij Qj k = Qik for any .i, j, k = 1, 2, 3. This means that no matter how many interactions occur, the multiplication factor is always an element of the matrix and thus is uniformly bounded. Therefore, the diverge model with fixed turning percentages is well-posed. However, without .α1,2 and .α1,3 being constants, this is no longer true as we have shown in the counter example in Sect. 8.5.1.1. The wellposedness requires some additional sufficient condition to hold, and the proof needs more elaborate argument that takes into account the .μ-waves. These will be done in Theorem 8.2.

8.5.1.3

A Sufficient Condition for the Well-Posedness of the Diverge Model

As we have previously demonstrated, the diverge model with time-varying vehicle turning percentages may not depend continuously on its initial data, which are defined in terms of the two-tuple .(ρ, μ). In this section, we propose an additional condition which guarantees continuous dependence with respect to the initial data at the diverge junction. Our analysis relies on the method of wave-front tracking (Sect. 8.1.2) and the technique of generalized tangent vectors (Sect. 8.5.1.2). Theorem 8.2 (Well-Posedness of the Diverge Junction) Consider the diverge junction (Fig. 8.3) and assume that 1. there exists some .δ > 0 such that the supplies .S2 (t) ≥ δ, .S3 (t) ≥ δ for all t; and 2. the path disaggregation variable .μ(t, x) has bounded total variation in t Then the solution of the diverge junction depends continuously on the initial and boundary conditions in terms of .ρ and .μ. Proof This proof is completed in several steps.

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8 Dynamic Network Loading: Physical Queue Models

Step 1. We invoke the wave-front tracking framework and the tangent vector technique to show the desired continuous dependence. As previously discussed, it boils down to showing that the increase in the tangent vector norm, as a result of an arbitrary number of wave interactions (including both .ρ-wave and .μ-wave), remains bounded. Step 2. First of all, the end of Sect. 8.5.1.2 shows that, for constant values of .μ (i.e., with fixed vehicle turning ratios), the tangent vector norm is uniformly bounded regardless of the interactions between the .ρ-waves and the junction. Next, one needs to consider the case where .μ changes from one value to another, i.e., when a .μ-wave interacts with the junction. In general, consider two consecutive interaction times of the .μ-waves, .tk and .tk+1 , and let .βk be the multiplication factor of the tangent vector norm between these two times. More precisely, we have .v(tk+1 −) = βk v(tk ), where .v(t) is the tangent vector norm at time t. Clearly, .βk can only take values in k and .α k are given by the constant .μ value during the below matrix where .α1,2 1,3 .[tk , tk+1 ]; see (8.115). ⎤ k k 1 α1,2 α1,3 . ⎢ k k k /α k ⎥ , .Q = {Qkij } = ⎣ 1/α1,2 1 α1,3 1,2 ⎦ k α k /α k 1/α1,3 1 1,2 1,3 ⎡

k = 0, 1, 2, . . . (8.116)

A similar argument may be used for all other intervals .[tk+1 , tk+2 ], .[tk+2 , tk+3 ] and so on. Our goal for the rest of this proof is to estimate the multiplication factor for the tangent norm under the WFT framework, assuming an arbitrary interaction pattern of the .ρ- and .μ-waves. . Step 3. We define .γ = Cδ1 > 0 where .δ is stated in the hypothesis of this theorem and .C1 is the flow capacity of link 1. Without loss of generality, we let .γ < 12 . Throughout Step 3, we will assume that .α1,2 (t) ≥ γ and .α1,3 (t) ≥ γ . As we demonstrated at the end of Sect. 8.5.1.2, in order to estimate the increase in the tangent vector norms, it suffices to consider the following type of sequence in which waves interact with the junction: .i → j then .j → k; in other words, the recipient of the previous .ρ-wave interaction is the source of the next .ρ-wave interaction. Such a chain of events is illustrated in Fig. 8.13. -wave interaction Last -wave interaction I (k )

tk

I (k+1)

tk+2

tk+1 k 1,2

,

k 1,3

Fig. 8.13 Illustration of .I (k)

I (k+2)

k+1 1,2

,

k+1 1,3

tk+3 k+2 1,2

,

k+2 1,3

t

8.5 Continuity of the Effective Delay Operator for LWR-Based Dynamic. . .

337

For each .k ≥ 1, we let .I (k) ∈ {1, 2, 3} be the recipient of the last interacting wave during .[tk , tk+1 ); see Fig. 8.13. The .tk represent the times when .μ-waves interact with the junction, changing the turning ratios .α1,2 and .α1,3 . Crosses represent the times at which .ρ-waves interact. The circles represent the last .ρ-wave within the interval during which .μ is constant. .I (k) denotes the recipient of the last interacting .ρ-wave. This gives rise to the sequence .{I (k) }. We make the following crucial observation. Consider any three consecutive elements in the sequence of the form .I (k) = i, .I (k+1) = 2 and .I (k+2) = j where .i, j ∈ {1, 2, 3}. By definition, the product of the multiplication factors within .[tk+1 , tk+3 ] is  k+1 .Q i2

· Qk+2 2j

=

Ak+1 2

· Qlij

Ak+1 2

where

∈ 1,

k+1 α1,2 k+2 α1,2

,

k+1 α1,2 k+2 α1,2

·

k+2 α1,3



k+1 α1,3

for .l ∈ {k +1 , k +2} where we use the superscripts to indicate the association of the variables to a specific time interval. The significance of this expression is that the multiplication factor can be decomposed into a term .A2 with a very specific structure (to be further elaborated below), and a term that would have been the multiplication factor as if the middle link .2 = I (k+1) was removed. Clearly, the above argument applies equally to the case where .I (k+1) = 3, and we use  k+1 .A 3

∈ 1,

k+1 α1,3 k+2 α1,3

,

k+1 α1,3 k+2 α1,3

·

k+2 α1,2



k+1 α1,2

to represent the term factored out if the middle link .I (k+1) = 3 is removed. By repeating this procedure, one may eliminate links 2 and 3 from the sequence (k) } except when they are the first or the last in this finite sequence. As a result, .{I k+1 the multiplicative terms .{Ak+1 2 } and .{A3 } are factored out. Therefore, the entire multiplication factor for the tangent norm is bounded by ∞ .

A2km +1

m=1

·

∞ n=1

A3kn +1 ·

1 γ2

(8.117)

∞ (km ) = 2 and .I (kn ) = where .{km }∞ m=1 and .{kn }n=1 are subsequences of .{k} such that .I 3. The first two terms in (8.117) are results of eliminating 2 and 3 from the sequence (k) }; the third term takes care of the first element and possibly the last element .{I in the sequence and is derived from the observation that .Qkij ≤ γ1 for all .i, j ∈ {1, 2, 3} and .k ≥ 0. It remains to show that the first (and second) term of (8.117) is bounded, and hence (8.117) is bounded. We write, without referring explicitly to the multiplication indices, that ∞ .

m=1

A2km +1

k+1 k+2 α1,2 α1,3 · = k+2 k+1 α1,2 α1,3 k k

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8 Dynamic Network Loading: Physical Queue Models

The above expression yields ∞ .

A2km +1

= exp

$ 

% k+1 log α1,2

k+2 − log α1,2

k

m=1

· exp

$ 

$

% k+2 k+1 log(1 − α1,2 ) − log(1 − α1,2 )

k

%  1   k+1 k+2  α1,2 − α1,2 · ≤ exp sup α∈[γ , 1−γ ] |α| k $ %   1 k+2 k+1 α  · exp sup · 1,2 − α1,2 α∈[γ , 1−γ ] |1 − α| k     1 1 ≤ exp · T V (μ) · exp · T V (μ) < ∞ γ γ where .T V (μ) is the bounded total variation of the path disaggregation variable .μ. Step 4. In this part of the proof, we deal with the situation where .α1,2 (t) < γ ; the other case where .α1,3 (t) < γ is entirely similar. We begin with the following observation:     S2 (t) S3 (t) S3 (t) out .f1 (t) = min D1 (t), , = min D1 (t) , α1,2 (t) α1,3 (t) α1,3 (t) 2 (t) since . αS1,2 (t) >

S2 (t) S2 (t)C1 ≥ C1 ≥ D1 (t). In other words, the minimum is never γ = δ S2 (t) attained at . α1,2 (t) , and any .ρ-wave interacting from link 2 will not generate new waves on link 1 or 3 since .f1out (t) does not change before and after the interaction.

So the only interactions that may change the tangent vector norm are .1 → 3 and 3 → 1, and the corresponding multiplication factors are, respectively, .α1,3 (t) and 1 . α1,3 (t) . Consider any sequence of interacting times of the .μ-wave .{tk }. Since the interactions .1 → 1 or .3 → 3 do not generate any increase in the tangent norm, we only need to consider the following sequence of interactions: .. . . → 1 → 3 → 1 → 3 . . .. Consequently, the resulting multiplication factor is bounded by .

.



k α1,3 k

k+1 α1,3

·

1 1−γ

2

where the second multiplicative term deals with the first and the last element in k , 1 } = 1 < 1 for all k. We may then proceed the sequence since .max{α1,3 k k 1−γ α1,3

α1,3

as in Step 3 and conclude that this multiplication factor is bounded, provided that .T V (μ) < ∞. 

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339

The two sufficient conditions for the well-posedness of the diverge model will be further analyzed and substantiated in the next sections. In particular, we will show in Sect. 8.5.2 that the supply anywhere in the network is indeed bounded from below. The bounded variation of the path disaggregation variables will be interpreted in Sect. 8.5.3.1.

8.5.2 An Estimation of Minimum Network Supply This section provides a lower bound on the supply, which is a function of density, anywhere in the network during the entire time horizon .[0, T ]. Our finding is that the jam density or gridlock can never occur within any finite time horizon, and the supply anywhere in the network is bounded away from zero. Let .D be the set of destinations in the network. Without loss of generality, we assume that every destination .d ∈ D is incident to a virtual link that connects d to the rest of the network; see Fig. 8.14. For each .d ∈ D, we introduce the supply of this destination, denoted by .S d , as the maximum rate at which cars can be discharged from the virtual link connected to this destination. Effectively, there exists a bottleneck (see Fig. 8.14) serving as a buffer between the destination and the network. And, the supply of the destination is equal to the flow capacity of this bottleneck. Notice that in some literature such a bottleneck is completely ignored and the destination is simply treated as a sink with infinite receiving capacity. This is of course a special case of ours once we set the supply .S d to be infinity. However, such a supply may be finite and even quite limited under some circumstances due to, for example, ramp metering, limited parking spaces, or the fact that the destination is an aggregated subnetwork that is congested. We introduce a few more concepts and notations. L = min Li

.

i∈A˜

C min = min Ci

.

links

i∈A˜

: the minimum link length in the network, including virtual links : the minimum link flow capacity in the network, including virtual

Bottleneck

Network Virtual link

d

Fig. 8.14 Illustration of the receiving capacity (supply) of the destination

Destination

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8 Dynamic Network Loading: Physical Queue Models

    jam λ = max fi (ρi −) : the maximum backward wave speed in the network i∈A˜   .p ¯ = min pJ , 1 − pJ > 0 , where .pJ and .1 − pJ are the two priority .

J ∈VM

parameters for the two incoming links of a merge junction J , .VM denotes the set of merge junctions in the network D = min S d > 0 : the minimum supply of all the destination nodes .δ d∈D

δk = min

.

inf Si

inf  i∈A˜ t∈ k L , (k+1) L x∈[ai , bi ] λ λ

(ρi (t, x))

: the minimum supply at any loca-

 tion in the network during time interval . k Lλ , (k + 1) Lλ , where .k = 0, 1, . . . Theorem 8.3 Consider a network consisting of only merge and diverge junctions as shown in Fig. 8.3. Given any vector of path departure rates, the dynamic network loading procedure described by the PDAE system yields the following property in the solution   ∀k = 0, 1, . . . .δk ≥ p ¯ k min δ D , pC ¯ min (8.118) Proof Since .δk is defined in terms of the supplies, it suffices for us to focus only on those densities such that .ρi (t, x) > ρic for .i ∈ A, .(t, x) ∈ [0, T ] × [ai , bi ]. It is also useful to keep in mind that densities beyond the critical density always propagate backward in space. The  is divided into several steps. proof Step 1. .k = 0. We have .t ∈ 0, Lλ . Since the network is initially empty, all the supplies .Si (ρi (t, x)), .i ∈ A, (t, x) ∈ {0} × [ai , bi ], are maximal and equal to the respective flow capacities at .t = 0. Afterward, a higher-than-critical density or a lower-than-maximum supply can only emerge from the downstream end of a link and propagate backward along this link. Moreover,  these backward waves can never reach the upstream end of the link within . 0, Lλ since . Lλ is the minimum link traversal time of backward waves. A higher-than-critical density (backward wave) can only emerge in one of the following cases. Case (1). A forward wave from link 1 (see Fig. 8.3) interacts with the diverge junction and creates a backward wave in the same link. Case (2). A forward wave from .I4 (or .I5 ) interacts with the merge junction and creates a backward wave in .I4 or .I5 . Case (3). A forward wave interacts with a destination from the relevant virtual link and creates a backward wave in the same virtual link. Each individual case will be investigated in detail below. • Case (1). We use the same notation shown in Fig. 8.3.  According to the reason provided above, .Si (t) ≡ Ci , .i = 2, 3 for .t ∈ 0, Lλ . Let the time of interaction

8.5 Continuity of the Effective Delay Operator for LWR-Based Dynamic. . .

341

be .t¯. Recall from (8.11) that  f1out (t¯+) = min D1 (t¯),

.

C3 C2 , α1,2 (t¯) α1,3 (t¯)

 (8.119)

If the minimum is attained at .D1 (t¯), then the entrance of link 1 will remain in the uncongested phase, i.e., .ρ1 (t+, b1 −) ≤ ρ1c after the interaction. Hence, no lower-than-maximum supply is generated. On the other hand, if say . α C2(t¯) is the 1,2 smallest, then S1 (ρ1 (t¯+, b1 −)) ≥ f1 (ρ1 (t¯+, b1 −)) = fout,1 (t¯+) =

.

C2 α1,2 (t¯+)

≥ C2 ≥ C min

(8.120)

In summary, the supply values corresponding to the backward waves generated at link 1, if any, are uniformly above .C min . • Case (2). Weturn to the merge junction shown in Fig. 8.3, and note .S6 (t) ≡ C6 for .t ∈ 0, Lλ . As usual, we let .t¯ be the time of interaction. There are two further cases for the merge junction, as shown in Fig. 8.4. We first consider the situation illustrated in Fig. 8.4a. Clearly, we have that S4 (ρ4 (t¯+, b4 −)) ≥ f4 (ρ4 (t¯+, b4 −)) = f4out (t¯+) = pC6 ≥ pC ¯ min.

.

(8.121) S5 (ρ5 (t¯+, b5 −)) ≥ f5 (ρ5 (t¯+, b5 −)) = f5out (t¯+) = (1 − p)C6 ≥ pC ¯ min (8.122) That is, the supply values of the backward waves generated at the downstream ends of .I4 and .I5 are uniformly above .pC ¯ min . For the situation depicted in Fig. 8.4b, we first note that the coordinates of out out  .Q are .(pC6 , (1 − p)C6 ) and the coordinates of the solution .(f 4 , f5 ) either satisfy f4out < pC6

.

and

f5out > (1 − p)C6

(8.123)

f5out < (1 − p)C6

(8.124)

as shown exactly in Fig. 8.4b or f4out > pC6

.

and

Taking (8.123) as an example (the (8.124) case can be treated similarly), we have S5 (ρ5 (t¯+, b5 −)) ≥ f5 (ρ5 (t¯+, b5 −)) > (1 − p)C6 ≥ pC ¯ min ,

.

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8 Dynamic Network Loading: Physical Queue Models

and no increase in density is produced at the downstream end of link .I4 as its exit flow is equal to the demand. To sum up, the supply values corresponding to the backward waves generated at .I4 and .I5 , if any, are uniformly above .pC ¯ min . • Case (3). For each destination .d ∈ D, let .S d be its supply and denote by vl the virtual link connecting to this destination. If .S d ≥ Cvl , then no backward waves can be generated on this link, and the supply value on the link is always equal to .Cvl . If .S d < Cvl , then only one higher-than-critical density can exist on this c and .f (ρ) = S d . Clearly, its supply virtual link, that is, .ρ such that .ρ > ρvl vl d value is equal to .S . In summary, the supply value corresponding to the backward waves generated at any virtual link, if any, is bounded below by .δ D . Finally, we notice that all the backward waves exhaustively described above originate from the downstream end of a link, and they cannot reach the upstream end of the same link within period . 0, Lλ . Thus, these backward waves cannot bring further reduction in the supply values by means of interacting with the junctions. We conclude that during period . 0, Lλ , the minimum supply in the network, .δ0 , is bounded below by   δ0 ≥ min δ D , pC ¯ min

.

(8.125)

Step 2. We move on to .k ≥ 1. In addition to Cases (1)–(3), which do not bring any supply values below .min{δ D , pC ¯ min }, two more cases may arise in which higherthan-critical densities may be generated as a result of a backward wave interacting with a junction: Case (4). A backward wave from link 2 (or 3, see Fig. 8.3) interacts with the diverge junction and creates a backward wave in link 1. Case (5). A backward wave from .I6 (see Fig. 8.3) interacts with the merge junction and creates a backward wave in .I4 or .I5 . Cases (4) and (5) will be analyzed in detail below. • Case (4). Without loss of generality, we assume the backward wave that interacts with the diverge junction is coming from link 2 and has the density value jam − c ¯ .ρ 2 ∈ (ρ2 , ρ2 ]. Let .t be the time of this interaction. In view of (8.119), if the minimum is attained at .D1 (t¯), then the interaction does not bring any increase in density at the downstream end of link 1, hence no decrease in the supply there. ¯ If the minimum is attained at . αS2 ((t t)¯) , we deduce in a similar way as (8.120) 1,2 that S1 (ρ1 (t¯+, b1 −)) ≥ f1 (ρ1 (t¯+, b1 −)) = fout,1 (t¯+)

.

=

S2 (ρ2− ) ≥ S2 (ρ2− ) ≥ δk−1 α1,2 (t¯+)

(8.126)

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343

The last inequality is due to the fact that a backward wave such as .ρ2− must be created at the downstream end of link 2 at a time earlier than .k Lλ ; thus its supply value .S2 (ρ2− ) must be bounded below by .δk−1 . In summary, the supply values corresponding to the backward waves generated at link 1, if any, are bounded below by .δk−1 . • Case (5). For the merge junction, we begin with the case illustrated in Fig. 8.4a. Assume the backward wave that interacts with the merge junction from .I6 has jam the density value .ρ6− ∈ (ρ6c , ρ6 ]. Similar to (8.121)–(8.122), we have S4 (ρ4 (t¯+, b4 −)) ≥ f4 (ρ4 (t¯+, b4 −)) = f4out (t¯+) = pS6 (ρ6− ) ≥ pδ ¯ k−1. (8.127)

.

S5 (ρ5 (t¯+, b5 −)) ≥ f5 (ρ5 (t¯+, b5 −)) = f5out (t¯+) = (1 − p)S6 (ρ6− ) ≥ pδ ¯ k−1 (8.128) where the last inequalities are due to the same reason provided in Case (4). In summary, the supply values corresponding to the backward waves generated at .I4 or .I5 , if any, must be bounded below by .pδ ¯ k−1 .  L L So far, we have shown that for .t ∈ k λ , (k + 1) λ , the presence of higher-thancritical densities, as exhaustively illustrated through Case (1)–(5), brings supply values throughout the entire network that are bounded below by

.

    ¯ k−1 min min δ D , pC ¯ min , δk−1 , pδ &'() & '( ) & '( ) Case (1)-(3)

Case (4)

Case(5)



= min δ D , pC ¯ min , pδ ¯ k−1



  = min δ D , pδ ¯ k−1

(8.129)

Step 3. Recall (8.125) and (8.129). That is   δ0 ≥ min δ D , pC ¯ min ,

.

  δk ≥ min δ D , pδ ¯ k−1

∀k ≥ 1 (8.130)

    . . We define .δˆ0 = min δ D , pC ¯ min and .δˆk = min δ D , p¯ δˆk−1 , .k ≥ 1. Clearly, ˆk ≤ δk for all .k ≥ 0. Furthermore, we deduce .δ ∀k ≥ 1, δˆ0 ≤ δ D ⇒ p¯ δˆk−1 ≤ δ D   which implies that .δˆk = min δ D , p¯ δˆk−1 = p¯ δˆk−1 , .k ≥ 1. Thus .

  δk ≥ δˆk = pδ ¯ k−1 = p¯ k δˆ0 = p¯ k min δ D , pC ¯ min

.

∀k ≥ 0 

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8 Dynamic Network Loading: Physical Queue Models

Theorem 8.3 guarantees that the supply values during each period . k Lλ , (k+1) Lλ anywhere in the network are uniformly bounded below by some constant that depends only on k, although such a constant may decay exponentially as k increases. Given that any dynamic network loading problem is conceived in a finite time horizon, we immediately obtain a lower bound on the supplies, as shown in the corollary below. Corollary 8.1 Under the same setup as Theorem 8.3, we have .

min

inf

i∈A x∈[ai , bi ]

     t  Si ρi (t, x) ≥ p¯ L/λ δ D , pC ¯ min

∀t ∈ [0, T ]

(8.131)

and .

min

inf

    Tλ inf Si ρi (t, x) ≥ p¯  L  min δ D , pC ¯ min

i∈A x∈[ai , bi ] t∈[0, T ]

(8.132)

where .· means rounding to the nearest integer from below. Remark 8.4 Corollary 8.1 proves that for any network consisting of the merge and diverge junctions and the junction models considered in this chapter, the supply values are uniformly bounded from below for any point in the spatial-temporal domain. In particular, a jam density can never occur in this network. Moreover, such a network is free of complete gridlock.4

8.5.3 The Continuity Result 8.5.3.1

Estimates Regarding the Path Disaggregation Variable

In this section we establish some properties of the path disaggregation variable .μ, which will verify the second hypothesis of Theorem 8.2. Lemma 8.2 Assume that there exists .M > 0 and . > 0 so that the following hold: 1. For all links i, the fundamental diagram .fi (·) is uniformly linear near zero density; more precisely, .fi (ρ) is constant on the interval .[0, ] for every i. jam 2. Each .fi (·) has a nonvanishing derivative, i.e., .|fi (ρ)| ≥ , ∀ρ ∈ [0, ρi ]. 3. The departure rates .hp (·), p ∈ P are uniformly bounded, have bounded variation, and are bounded away from zero when they are non-zero; i.e., .T V (hp ) < M and .hp (t) ∈ {0} ∪ [ , M] for every p.

4 A complete gridlock refers to the situations where a non-zero static solution of the PDAE system exists. Intuitively, it means that a complete jam .ρ = ρ jam is formed somewhere in the network and the static queues do not dissipate in finite time.

8.5 Continuity of the Effective Delay Operator for LWR-Based Dynamic. . .

345

Then the path disaggregation variables .μ(t, x) are either zero or uniformly bounded away from zero. Moreover, they have bounded variation. Proof The proof is divided into a few steps. Step 1. Notice that assumption 3 implies that the departure rates .hp are non-zero on a finite set of intervals. Indeed, let n be the number of intervals where .hp does not vanish; then .n ≤ T V (hp ) < ∞, which implies that n is finite. Step 2. Using the assumption that each fundamental diagram .fi (·) has a nonvanishing derivative, we have, at each origin, that bounded total variation of the density must imply bounded total variation of the flow, and vice versa. Therefore, the assumption on the bounded variation of .hp implies that the density and path disaggregation variable are of bounded variation on the virtual link incident to that origin. Step 3. Assumption 1 guarantees that, on each link, all waves of the type .(0, ρ) or .(ρ, 0) are contact discontinuities for .ρ sufficiently small and, in particular, travel with a constant speed. Moreover, all .μ-waves travel with the same speed for low densities. Therefore, taking into account Step 1, whenever .μ is non-zero, it is uniformly bounded away from zero on the entire network. In particular there exists p   . > 0 so that .μ (t, x) ∈ {0} ∪ [ , 1] for every .t, x, i and p. Therefore, at diverging i junctions, the coefficients .α1,2 (t) and .α1,3 (t) satisfy the same properties. Step 4. Let us now turn to the total variation of the path disaggregation variables. We know, from Step 2, that .μ has bounded variation on virtual links incident to the origins. We also know from Step 3 that .μ is bounded away from zero whenever it is non-zero, and the same holds for the turning percentages at diverge junctions. Consider a .μ-wave .(μl , μr ); then its variation .|μl − μr | can change − only upon interaction with diverge junctions. More precisely, denote by .(μ− l , μr ) + and .(μl , μ+ r ) the wave before and after the interaction, respectively; we have + + | ≤ 1 |μ− − μ− | and bounded away from zero, where .α = α .|μ − μ 1,2 or r r l α l .α = α1,3 . Since the .μ-waves travel only forward on the links with uniformly bounded speed, we must have that the interactions with diverge junctions can occur only a finite number of times. Thus we conclude that .μ has bounded total variation.  Remark 8.5 Notice that assumptions 1 and 2 of Lemma 8.2 are satisfied by the Newell-Daganzo (triangular) fundamental diagrams, where both the free flow and congested branches of the FD are linear. Moreover, given an arbitrary fundamental diagram, one can always make minimum modifications at .ρ = 0 and .ρ = ρ c to comply with conditions 1 and 2. Assumption 3 of Lemma 8.2 is satisfied by any departure rate that results from a finite number of cars entering the network. And, again, any departure rate can be adjusted to satisfy this condition with minor (in fact, as small as one wants) modifications.

8.5.3.2

Well-Posedness of the Queuing Model at the Origin with Respect to Departure Rates

In this section we discuss the continuous dependence of the queues .qs (t) and the p solutions .ρi and .μi with respect to the path departure rates .hp (t), .p ∈ P.

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8 Dynamic Network Loading: Physical Queue Models

Following Herty et al. (2007), we introduce a generalized tangent vector i,p p (ηs , ξρi , ξμ ) to the triplet .(qs , ρi , μi ) where .ηS ∈ R is a scalar shift of the queue .qS , p i.e., the shifted queue is .qs + ηs , while the tangent vectors to .ρi and .μi are defined as before in Sect. 8.5.1.2. The tangent vector norm of .ηs is simply its absolute value .|ηs |. .

Lemma 8.3 Assume that the departure rates .hp (·) are piecewise constant, and let ξp be a tangent vector defined via shifting the jumps in .hp (·). Then the tangent i,p vector .(ηs , ξρi , ξμ ) is well defined; its norm is equal to that of .ξp and bounded for all times.

.

j

Proof Let .ξp be the shift of the j -th jump of .hp that occurs at time .tj . Then the j j equation of .qs is possibly affected on the interval .[tj , tj + ξp ] (assuming .ξp > 0). More precisely, if .qs (t¯) > 0 then no wave is generated on the virtual link, while j a shift in the queue is generated with .ηs = Δj hp · ξp , where .Δj hp is the jump in .hp occurring at time .tj . On the other hand, if .qs (t¯) = 0, then a wave .(ρ − , ρ + ) is produced at time .tj on j j the virtual link with shift .ξρ = λ · ξp and .λ is the speed of the wave .(ρ − , ρ + ). We can then compute ξρj · (ρ + − ρ − ) = ξp · λ · (ρ + − ρ − ) = ξp · j

.

j

f (ρ + ) − f (ρ− ) · (ρ + − ρ − ). (ρ + − ρ − )

Moreover, .f (ρ + ) − f (ρ − ) = Δj hp ; thus the norm of the tangent vector generated is the same as before. i,p To prove that the norm of the tangent vector .(ηs , ξρi , ξμ ) is bounded for all times, let us first consider a wave .(ρ − , ρ + ) with negative speed interacting with the j queue .qs with shift .ξρ . Then we can write q˙s+ − q˙s− = f (ρ − ) − f (ρ + )

.

where .q˙s− and .q˙s+ are the time derivatives of .qs before and after the interaction, respectively. Therefore, letting .λ be the speed of the wave .(ρ − , ρ + ), we get ξρ ξρ · (ρ + − ρ − ) ·(f (ρ − )−f (ρ + )) = ·(f (ρ − )−f (ρ + )) = ξρj ·(ρ + −ρ − ); λ f (ρ − ) − f (ρ + ) j

ηs =

.

j

thus the norm of the tangent vector is bounded. i,p The norm of the tangent vector .(ηs , ξρi , ξμ ) may also change due to emptying  of the queue .qs , but this can be treated as in Herty et al. (2007).

8.5 Continuity of the Effective Delay Operator for LWR-Based Dynamic. . .

8.5.3.3

347

The Continuity of the Path Delay Operator

We are able to present the continuity result for the delay operator, when one employs the continuous-time LWR model with spillback incorporated. Theorem 8.4 (Continuity of the Delay Operator) Consider a network consisting of diverge and merge junctions described in Sects. 8.1.1.2 and 8.1.1.3. Under the same assumptions stated in Lemma 8.2, the path delay operator, as a result of the dynamic network loading model presented in Sect. 8.1.3, is continuous. Proof We have shown that at each node, either an origin, diverge node, or merge node, the solution depends continuously on the initial and boundary values. In addition, between any two distinct nodes, the propagation speed of either the .ρor the .μ-wave is uniformly bounded. Thus, well-posedness holds on the level of the entire network. Consequently, the vehicle travel speed .vi (ρi ) for any i also depends continuously on the departure rates. We thus conclude that the path travel  time depends continuously on the departure rates. The assumption that the network consists of only merge and diverge nodes is not restrictive since junctions with general topology can be decomposed into a set of elementary junctions such as the merge and the diverge junctions—a process illustrated in Daganzo (1995). In addition, junctions that are also origins/destinations can be treated in a similar way by introducing virtual links. Remark 8.6 Szeto (2003) provides an example of discontinuous dependence of the travel time on the path departure rates using the cell transmission model representation of a signal-controlled network. In particular, the author showed that when a queue generated by the red signal spills back into the upstream junction, the experienced path travel time jumps from one value to another. This, however, does not contradict our result presented here for the following reason: the jam density caused by the red signal in Szeto (2003) does not exist in our network, which has only merge and diverge junctions (without any signal controls). Indeed, as shown in Theorem 8.3, the supply functions at any location in the network are uniformly bounded below by a positive constant, and thus the jam density will never occur in a finite time horizon. The reader is reminded of the example presented in Sect. 8.5.1.1, where the illposedness of the diverge model is caused precisely by the presence of a jam density. The counterexample from Szeto (2003) is constructed essentially in the same way as our example, by using signal controls that create the jam density value. Here, we would like to comment on the assumptions made along the way in proving the continuity of the delay operator. In a recent paper by Bressan and Yu (2015), a counterexample of uniqueness and continuous dependence of solutions is provided under certain conditions. More precisely, the authors construct the p counterexample by assuming that the path disaggregation variables .μi have infinite total variation. This shows the necessity of the second assumption of Theorem 8.2. Moreover, again in Bressan and Yu (2015), the authors prove that for density

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8 Dynamic Network Loading: Physical Queue Models

oscillating near zero, the solution may not be unique (even in the case of bounded total variation), which shows the necessity of the third assumption in Lemma 8.2.

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Chapter 9

Numerical Results

9.1 Closed-Form Solutions of Dynamic User Equilibria In this section we present closed-form solutions of DUEs with fixed demand (Sect. 9.1.1) and elastic demand (Sect. 9.1.3) in a simple network depicted in Fig. 9.1. The network has three links and three nodes, with two origin-destination (O-D) pairs: W =

.



(1, 3), (2, 3)



There are three paths connecting the two O-D pairs. Following the notation of Chaps. 1 and 3, we have P1,3 = {p1 , p2 }

.

P2,3 = {p3 } p1 = {a1 } p2 = {a2 , a3 } p3 = {a3 } We employ the Vickrey model (Sect. 7.2) to govern the arc dynamics. The Vickrey model assumes that vehicles entering a link will travel at free flow speed for a fixed time before they reach the exit, where a bottleneck with finite flow capacity is located. A queue with zero spatial volume will form in front of the bottleneck when the flow of arriving vehicles exceeds the bottleneck capacity. The arc dynamics can

© Springer Nature Switzerland AG 2022 T. L. Friesz, K. Han, Dynamic Network User Equilibrium, Complex Networks and Dynamic Systems 5, https://doi.org/10.1007/978-3-031-25564-9_9

353

354

9 Numerical Results

Fig. 9.1 Three-arc, three-node traffic network

2 a3

a2

1

Table 9.1 Arc parameters

Arc name .a1 .a2 .a3

a1

Free flow time .t0 30 (min) 10 (min) 20 (min)

3

Bottleneck capacity M 60 (vehicle/min) 60 (vehicle/min) 120 (vehicle/min)

be described by an ordinary differential equation (ODE)  d M . q(t) = u(t − t0 ) −   dt min u(t − t0 ), M

if q(t) = 0 if q(t) = 0

(9.1)

where .q(t) denotes the size of the queue, .t0 denotes the free flow travel time, and M denotes the bottleneck capacity. Moreover, the link traversal time consists of the fixed free flow time and a congestion-related queuing time λ(t) = t0 +

.

q(t + t0 ) M

(9.2)

where .λ(t) denotes the time taken to traverse the link when entering the link at time t. A much more detailed discussion of the Vickrey model including a closed form solution to the ODE (9.1) will be presented later in Chap. 7. The arc-specific parameters are summarized in Table 9.1. p In what follows, we use .qa (t) to denote the queue on arc a that is associated with path p such that .a ∈ p. The time horizon is .[t0 , tf ] = [0, 120] (in minute). In order to articulate the departure time choice, the desired arrival times for O-D pairs .(1, 3) and .(2, 3) are set to be .T1,3 = T2,3 = 90. The asymmetric early/late arrival penalty functions for the O-D pairs are F1,3 [τ ] =

.

F2,3 [τ ] =

 −1/2τ 2τ  −1/4τ τ

τ ∈ (−∞, 0] τ ∈ [0, +∞) τ ∈ (−∞, 0] τ ∈ [0, +∞)

.

(9.3)

(9.4)

where .τ = t + Dp (t, h) − Ti,j indicates the deviation of actual arrival time from the desired arrival time, .(i, j ) = (1, 3) or .(2, 3).

9.1 Closed-Form Solutions of Dynamic User Equilibria

355

9.1.1 Simultaneous Route-and-Departure-Time DUE with Fixed Demand We assume fixed travel demands of .Q1,3 = 6000 and .Q2,3 = 3000. The equilibrium path departure rates (path flows) are given explicitly as follows. Note that these closed-form solutions were manually derived to illustrate the DUE notion (Figs. 9.2, 9.3 and 9.4). ⎧ ⎪ t ∈ [20, 40] ⎪ ⎨120 ∗ .hp (t) = (9.5) 20 t ∈ (40, 70] 1 ⎪ ⎪ ⎩0 otherwise

∗ .hp (t) 2

⎧ ⎪ ⎪ ⎨90

=

40 ⎪ ⎪ ⎩0 

∗ .hp (t) 3

=

t ∈ [20, 40] t ∈ (40, 70]

(9.6)

otherwise

100

t ∈ [30, 60]

0

otherwise

(9.7)

150

Path flow (veh/min)

120 90 60 30

0

20

40

Fig. 9.2 Equilibrium path flow along .p1

60 Time (min)

80

100

120

356

9 Numerical Results

150

Path flow (veh/min)

120 90 60 30

0

20

40

60 Time (min)

80

100

120

60 Time (min)

80

100

120

Fig. 9.3 Equilibrium path flow along .p2

150

Path flow (veh/min)

120 90 60 30

0

20

40

Fig. 9.4 Equilibrium path flow along .p3

9.1 Closed-Form Solutions of Dynamic User Equilibria

357

Using Eq. (9.1), we can easily compute the temporal trajectories of p p p p qa11 , qa22 , qa32 , qa33 as follows

.

p

qa11 (t) =

.

⎧ ⎪ ⎪ ⎨60(t − 50)

−40(t − 100) ⎪ ⎪ ⎩0

⎧ ⎪ ⎪ ⎨30(t − 30) p2 .qa2 (t) = −20(t − 80) ⎪ ⎪ ⎩0

p2 .qa3 (t)

=

⎧ ⎪ ⎪ ⎨15(t − 50)

−60(t − 100) ⎪ ⎪ ⎩0

⎧ ⎪ ⎪ ⎨25(t − 50) p3 .qa3 (t) = −75(t − 90) ⎪ ⎪ ⎩0

t ∈ [50, 70] t ∈ (70, 100]

(9.8)

otherwise t ∈ [30, 50] t ∈ (50, 80]

(9.9)

otherwise t ∈ [50, 90) t ∈ (90, 100]

(9.10)

otherwise t ∈ [50, 80] t ∈ (80, 90]

(9.11)

otherwise

Let us now compute the path delays .Dp1 (·, h∗ ), Dp2 (·, h∗ ), Dp3 (·, h∗ ). In view of (9.2), the total path traversal time function is calculated below. ⎧ ⎪ ⎪ ⎨t + 10 ∗ .Dp1 (t, h ) = −2/3t + 230/3 ⎪ ⎪ ⎩30

∗

Dp2 (t, h ) =

.

⎧ ⎪ ⎪ ⎨t + 10

−2/3t + 230/3 ⎪ ⎪ ⎩30

⎧ ⎪ ⎪ ⎨1/3 t + 10 ∗ .Dp3 (t, h ) = −1/2 t + 60 ⎪ ⎪ ⎩20

t ∈ [20, 40] t ∈ (40, 70]

(9.12)

otherwise t ∈ [20, 40] t ∈ (40, 70]

(9.13)

otherwise t ∈ [30, 60] t ∈ (60, 80] otherwise

(9.14)

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9 Numerical Results

We then readily verify from the definition of the effective path delays and (9.3) and (9.4) that ⎧ ⎪ t ∈ [0, 20] ⎪ ⎨−1/2 t + 60 ∗ .Ψp1 (t, h ) = (9.15) 50 t ∈ (20, 70] . ⎪ ⎪ ⎩2 t − 90 t ∈ (70, 120]

∗

Ψp2 (t, h ) =

⎧ ⎪ ⎪ ⎨−1/2 t + 60 50 ⎪ ⎪ ⎩2 t − 90

⎧ ⎪ ⎪ ⎨−1/4 t + ∗ Ψp3 (t, h ) = 30 ⎪ ⎪ ⎩t − 50

t ∈ [0, 20] t ∈ (20, 70]

(9.16)

.

t ∈ (70, 120]

75 2

t ∈ [0, 30] t ∈ (30, 80]

(9.17)

t ∈ (80, 120]

The equilibrium path flows and associated effective path delays are plotted in Figs. 9.5, 9.6 and 9.7. It is visibly clear that these plots indicate equilibrium solutions. effective delay

150

150

120

120

90

90

60

60

30

30

0

20

40

60 Time (min)

80

Fig. 9.5 Equilibrium path flow and effective delay along path .p1

100

120

unit effective delay

Path flow (veh/min)

path flow

9.1 Closed-Form Solutions of Dynamic User Equilibria

359

effective delay

150

150

120

120

90

90

60

60

30

30

0

20

40

60 Time (min)

80

100

unit effective delay

Path flow (veh/min)

path flow

120

Fig. 9.6 Equilibrium path flow and effective delay along path .p2

effective delay

150

150

120

120

90

90

60

60

30

30

0

20

40

60 Time (min)

80

100

unit effective delay

Path flow (veh/min)

path flow

120

Fig. 9.7 Equilibrium path flow and effective delay along path .p3

More rigorously, let us verify that .h∗p1 , h∗p2 , h∗p3 solve the variational inequality (9.18) find h∗ ∈ Λ such that .

3

i=1

0

120

Ψpi (t, h∗ ) · (hpi − h∗pi ) dt ≥ 0

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ∀h ∈ Λ ⎪ ⎭

  V I Ψ, Λ, [t0 , tf ] (9.18)

360

9 Numerical Results

where

   Λ = h = h 1 , h2 , h3 ≥ 0 :

120 

.

0



120 0

 hp1 + hp2 dt = 6000,

 hp3 dt = 3000

  |P | ⊆ L2+ t0 , tf

(9.19)

We note from (9.15)–(9.17) that      v13 = min essinf Ψp1 (t, h∗ ) : t ∈ [0, 120] , essinf Ψp2 (t, h∗ ) : t ∈ [0, 120]

.

= 50

  v23 = essinf Ψp1 (t, h∗ ) : t ∈ [0, 120] = 30 Therefore, for any .h ∈ Λ, 3



120

.

i=1

≥

0

0

120

Ψpi (t, h∗ ) · hpi (t) dt

  v13 · hp1 (t) + hp2 (t) dt +



120

= 50 0



120

0



hp1 (t) + hp2 (t) dt + 30

v23 · hp3 (t) dt

120 0

hp3 (t) dt

= 50 × 6000 + 30 × 3000 =

3

i=1

0

120

Ψpi (t, h∗ ) · h∗pi (t) dt

  This shows that .h∗ = h∗p1 , h∗p2 , h∗p3 is indeed a solution of VI (9.18).

9.1.2 Kuhn-Tucker Conditions for the Fixed Demand DUE Let us use the Kuhn-Tucker conditions derived in Sect. 3.5.1 to verify that .h∗ given in Sect. 9.1.1 is a solution of the variational inequality. Recall that the Kuhn-Tucker conditions are equivalently expressed as the linear system (3.64).

9.1 Closed-Form Solutions of Dynamic User Equilibria

361

To apply the KT conditions for finite-dimensional problems, we consider a discretization of time interval .[0, 120] (in minute) with step size .Δ = 10 (minutes). The discrete-time equilibrium path flows are as follows (without causing any confusion, the notations below all refer to discrete-time quantities).  T h∗p1 = 0, 0, 120, 120, 20, 20, 20, 0, 0, 0, 0, 0 ∈ R12 +.  T h∗p2 = 0, 0, 90, 90, 40, 40, 40, 0, 0, 0, 0, 0 ∈ R12 +.   T h∗p3 = 0, 0, 0, 100, 100, 100, 0, 0, 0, 0, 0, 0 ∈ R12 +

.

(9.20) (9.21) (9.22)

the corresponding effective path delays are  T Ψp1 (h∗ ) = 60, 55, 50, 50, 50, 50, 50, 50, 70, 90, 110, 130 ∈ R12 +.

.

(9.23)   T Ψp2 (h∗ ) = 60, 55, 50, 50, 50, 50, 50, 50, 70, 90, 110, 130 ∈ R12 +. (9.24) Ψp3 (h∗ ) =

 75

, 35,

2

T 65 , 30, 30, 30, 30, 30, 30, 40, 50, 60 ∈ R12 + 2 (9.25)

T  Then we let .Ψ (h∗ ) = [Ψp1 (h∗ )]T , [Ψp2 (h∗ )]T , [Ψp3 (h∗ )]T . The matrix A in system (3.64) is   u(1, 3) u(2, 3) ∈ R36×2

A =

.

where u(1, 3) =



.

u(2, 3) =

.



1, 1, . . . , 1, 1, 0 . . . , 0       24 one’s 12 zero’s 0, 0, . . . , 0, 0, 1 . . . , 1       24 zero’s 12 one’s

T

T

The resulting system of linear equations is easily solved. The dual variable .π associated with inequality constraints is displayed graphically in Fig. 9.8. By observing that .π ≥ 0, we conclude the discrete solution .h∗ is in fact a dynamic user equilibrium.

362

9 Numerical Results

100

Value

80

60

40

20

0

0

5

10

15

20

25

30

35

Index Fig. 9.8 SRDT DUE with fixed demand: the dual variable .π associated with inequality constraints in the variational inequality. The horizontal axis indicates the order at which each element appears in the vector .π ; the vertical axis indicates the value of the corresponding element

9.1.3 Simultaneous Route-and-Departure-Time DUE with Elastic Demand In the three-arc, three-node network shown in Fig. 9.1, there are two origindestination pairs W =

.



(1, 3), (2, 3)



We denote by .Q1,3 the travel demand from node 1 to node 3 and .Q2,3 the travel demand from node 2 to node 3. From now on, .Q1,3 and .Q2,3 are no longer fixed constants as they represent elastic demands. In fact, it is not difficult to show  of equilibria exist corresponding  that a family to different demand profiles .Q = Q1,3 , Q2,3 . In order to describe such a family of equilibria, we introduce two dependent parameters .α, β ∈ [0, 120] such that α > β,

.

α + β = 60

Let us consider the following parametrized path flows: ⎧ ⎪ ⎪ ⎨120 ∗ 20 .hp (t) = 1 ⎪ ⎪ ⎩ 0

t ∈ [α  − β, α]  t ∈ α, α + 32 β . otherwise

(9.26)

9.1 Closed-Form Solutions of Dynamic User Equilibria

⎧ ⎪ ⎪ ⎨90 ∗ hp2 (t) = 40 ⎪ ⎪ ⎩ 0 ⎧ ⎨100 h∗p3 (t) = ⎩0

363

t ∈ [α  − β, α]  t ∈ α, α + 32 β .

(9.27)

otherwise   t ∈ 10 + α − β, 10 + α + 12 β

(9.28)

otherwise

We readily calculate with the above path flows that ⎧ ⎪ − 30 − α + β)  ⎪ ⎨60(t  p1 −40 t − 30 − α − 32 β .qa1 (t) = ⎪ ⎪ ⎩ 0

t ∈ [30 + α − β, 30 + α]  t ∈ 30 + α, 30 + α + 32 β . otherwise (9.29)

⎧ ⎪ − 10 − α + β)  ⎪ ⎨30(t  p2 qa2 (t) = −20 t − 10 − α − 32 β ⎪ ⎪ ⎩ 0

t ∈ [10 + α − β, 10 + α]  t ∈ 10 + α, 10 + α + 32 β . otherwise (9.30)

⎧ ⎪ ⎪ ⎨15(t − 30 − α + β) p2 qa3 (t) = −60(t − 30 − α − 32 β) ⎪ ⎪ ⎩ 0

t ∈ [30 + α − β, 30 + α + β]  t ∈ 30 + α + β, 30 + α + 32 β . otherwise (9.31)

p

qa33 (t) =

−75(t − 30 − α − β)

  t ∈ 30 + α − β, 30 + α + 12 β   t ∈ 30 + α + 12 β, 30 + α + β

0

otherwise

⎧ ⎪ ⎪25(t − 30 − α + β) ⎨ ⎪ ⎪ ⎩

(9.32)

364

9 Numerical Results p

where .qaij (·) denotes the (point) queue on link .ai associated with path .pj . The path delays are computed as ⎧ ⎪ ⎪ ⎨30 + t − α + β ∗ 30 + β − 23 (t − α) .Dp1 (t, h ) = ⎪ ⎪ ⎩ 30

t ∈ [α − β, α]  t ∈ α, α + 32 β .

⎧ ⎪ ⎪ ⎨30 + t − α + β ∗ Dp2 (t, h ) = 30 + β − 23 (t − α) ⎪ ⎪ ⎩ 30

t ∈ [α − β, α]  t ∈ α, α + 32 β .

(9.33)

otherwise

(9.34)

otherwise

⎧ ⎪ 20 + 13 (t − 10 − α + β) ⎪ ⎨ Dp3 (t, h∗ ) = 20 − 12 (t − 10 − α − 32 β) ⎪ ⎪ ⎩ 20

  t ∈ 10 + α − β, 10 + α + 12 β   t ∈ 10 + α + 12 β, 10 + α + 32 β otherwise (9.35)

Consequently, the unit effective path delays read ⎧ 1 ⎪ ⎪ ⎨60 − 2 t ∗ .Ψp1 (t, h ) = 60 − 12 α + 12 β ⎪ ⎪ ⎩2t − 90

∗

Ψp2 (t, h ) =

⎧ 1 ⎪ ⎪ ⎨60 − 2 t

+ 60 − ⎪ ⎪ ⎩2t − 90 1 2α

t ∈ [0, α − β] t ∈ (α − β, α + 32 β] . t ∈ (α +

3 2 β,

(9.36)

120]

t ∈ [0, α − β] 1 2β

⎧ 75 ⎪ − 14 t ⎪ ⎨2 1 1 Ψp3 (t, h∗ ) = 35 − 4 α + 4 β ⎪ ⎪ ⎩t − 50

t ∈ (α − β, α + 32 β] . t ∈ (α +

3 2 β,

(9.37)

120]

t ∈ [0, 10 + α − β]  t ∈ 10 + α − β, 10 + α + 32 β   t ∈ 10 + α + 32 β, 120 (9.38)

9.1 Closed-Form Solutions of Dynamic User Equilibria

365

It is thus clear from (9.36)–(9.38) that .h∗p1 (t), h∗p2 (t) and .h∗p3 (t) given by (9.26)– (9.28) yield a dynamic user equilibrium with demand Q13 = 300β,

Q23 = 150β

.

(9.39)

Remark 9.1 The numerical example of dynamic user equilibrium provided in Sect. 9.1 is a special case of the example discussed here with .α = 40, β = 20. We now introduce the inverse demand function as input of the elastic demand DUE problem   Θ = Θ13 , Θ23 : R2+ → R2++

.

with 1 (Q13 − 7000). 100 1 (Q23 − 4000) = Θ23 [Q23 ] = − 100

v13 = Θ13 [Q13 ] = −

(9.40)

v23

(9.41)

.

where .v13 and .v23 denote the minimum effective travel delays of O-D pair .(1, 3) and .(2, 3), respectively. In order to identify a DUE with elastic demand as in Definition 3.2, it suffices to choose, from the family of user equilibria, the one satisfying (9.40) and (9.41). We notice from (9.36)–(9.38) that the essential infimum ∗ ∗ .v 13 and .v23 can be expressed in terms of .β as v13 = 30 + β,

.

1 v23 = 20 + β 2

In view of (9.39), we have the following relationships between the minimum effective delay and the travel demand 1 Q13 , 300

v13 = 30 +

.

v23 = 20 +

1 Q23 300

(9.42)

Figure 9.9 shows the lines representing (9.42) together with the inverse demand functions (9.40) and (9.41). Based on Fig. 9.9, the DUE that satisfies the inverse demand constraints is identified via the intersections of the straight lines; that is, .Q13 = 3000 and .Q23 = 1500 (i.e., .β = 10). We now claim that the DUE with elastic demand is given by (9.26)–(9.28) with .α = 50, β = 10, that is,

h∗p1 (t) =

.

⎧ ⎪ ⎪ ⎨120 20 ⎪ ⎪ ⎩0

t ∈ [40, 50] t ∈ (50, 65] . otherwise

(9.43)

366

9 Numerical Results

v13 Inverse demand curve 60 v13 = 30 +

1 300

Q13

40

20

0

2000

4000

Q 13

6000

v23

60 v23 = 30 +

40

1 300

Q23

20 Inverse demand curve 0

2000

4000

6000

Q 23

Fig. 9.9 The inverse demand functions .Θ1,3 [·], Θ2,3 [·] and the relation between minimum effective delay and travel demand given by (9.42)

h∗p2 (t)

=

⎧ ⎪ ⎪ ⎨90 40 ⎪ ⎪ ⎩0

t ∈ [40, 50] t ∈ (50, 65] . otherwise

(9.44)

9.1 Closed-Form Solutions of Dynamic User Equilibria

 100

h∗p3 (t) =

0

367

t ∈ [50, 65]

(9.45)

otherwise

with an equilibriated travel demand of Q∗13 = 3000,

Q∗23 = 1500

.

(9.46)

The corresponding effective path delays are ⎧ ⎪ ⎪60 − 12 t ⎨

∗

Ψp1 (t, h ) =

.

40 ⎪ ⎪ ⎩2t − 90

⎧ 1 ⎪ ⎪ ⎨60 − 2 t Ψp2 (t, h∗ ) = 40 ⎪ ⎪ ⎩2t − 90 ⎧ 75 1 ⎪ ⎪ ⎨ 2 − 4t

∗

Ψp3 (t, h ) =

25 ⎪ ⎪ ⎩t − 50

t ∈ [0, 40] t ∈ (40, 65]

.

(9.47)

.

(9.48)

t ∈ (65, 120]

t ∈ [0, 40] t ∈ (40, 65] t ∈ (65, 120]

t ∈ [0, 50] t ∈ (50, 75]

(9.49)

t ∈ (75, 120]

We thus have  ∗    ∗ v13 , v23 = (40, 25) = Θ13 [Q∗13 ], Θ23 [Q∗23 ]

.

  This proves our claim that the pair . h∗ , Q∗ given by (9.43)–(9.46) is a solution of the dynamic user equilibrium with  elastic demand. Next, let us verify that such . h∗ , Q∗ indeed satisfies the variational inequality stated in Theorem 3.2. In fact, we have 3



120

.

i=1

=

2 i=1

0 ∗ v13

  Ψpi (t, h∗ )h∗pi (t) dt − Θ13 [Q∗13 ]Q∗13 + Θ23 [Q∗23 ]Q∗23



120 0

∗ h∗pi (t) dt + v23

0

120

  h∗p3 (t) dt − Θ13 [Q∗13 ]Q∗13 + Θ23 [Q∗23 ]Q∗23

  ∗ ∗ =v13 Q∗13 + v23 Q∗23 − Θ13 [Q∗13 ]Q∗13 + Θ23 [Q∗23 ]Q∗23 =0

368

9 Numerical Results

   we have On the other hand, given any . h, Q ∈ Λ, 3



  Ψpi (t, h∗ )hpi (t) dt − Θ13 [Q∗13 ]Q13 + Θ23 [Q∗23 ]Q23

120

.

i=1 2

≥

0 ∗ v13



120

hpi (t) dt

0

i=1

∗ + v23



120

0

  hp3 (t) dt − Θ13 [Q∗13 ]Q13 + Θ23 [Q∗23 ]Q23

  ∗ ∗ =v13 Q13 + v23 Q23 − Θ13 [Q∗13 ]Q13 + Θ23 [Q∗23 ]Q23 =0 Therefore, 3



120

.

i=1

0

  Ψpi (t, h∗ ) hpi − h∗pi dt

     ≥ 0 − Θ13 [Q∗13 ] Q13 − Q∗13 + Θ23 [Q∗23 ] Q23 − Q∗23

(9.50)

9.1.4 Kuhn-Tucker Conditions for the Elastic Demand DUE In this section, the Kuhn-Tucker conditions (3.70)–(3.73) and the equivalent linear system (3.79) will be used to verify that .(h∗ , Q∗ ) given in Sect. 9.1.3 is a dynamic user equilibrium with elastic travel demand. We consider a time-discretization of the interval .[0, 120] (in minute) with step size .Δ = 5 (minutes). Thus the discrete-time equilibrium path flows are as follows (without causing any confusion, the notations below all refer to discrete-time quantities). h∗p1 = (0, 0, 0, 0, 0, 0, 0, 0, 120, 120, 20, 20,

.

20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)T ∈ R24 +. h∗p2

(9.51)

= (0, 0, 0, 0, 0, 0, 0, 0, 90, 90, 40, 40, 40, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)T ∈ R24 +.

(9.52)

h∗p3 = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 100 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)T ∈ R24 +

(9.53)

9.1 Closed-Form Solutions of Dynamic User Equilibria

369

The corresponding discrete-time effective path delays are Ψp1 (h∗ ) =(60, 57.5, 55, 52.5, 50, 47.5, 45, 42.5, 40, 40, 40, 40,

.

40, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140)T ∈ R24 +.

(9.54)

∗

Ψp2 (h ) =((60, 57.5, 55, 52.5, 50, 47.5, 45, 42.5, 40, 40, 40, 40, 40, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140)T ∈ R24 +.

(9.55)

Ψp3 (h∗ ) =(37.5, 36.25 35, 33.75, 32.5, 31.25, 30, 28.75, 27.5, 26.25, 25, 25, 25, 25, 25, 25, 30, 35, 40, 45, 50, 55, 60, 65)T ∈ R24 +

(9.56)

We then form .Ψ (h∗ ) by concatenating the three path effective delays, that is, T  ∗ .Ψ (h ) = [Ψp1 (h∗ )]T , [Ψp2 (h∗ )]T , [Ψp2 (h∗ )]T . The matrix .Ae from linear system (3.79) becomes   Ae = u(1, 3) u(2, 3) ∈ R74×2

(1, 3), (2, 3) ∈ W

.

where

u(2, 3)



T 1, 1, . . . , 1, 0, 0, . . . , 0, −0.2, 0 .       48 one’s 24 zero’s  T = 0, 0, . . . , 0, 1, 1, . . . , 1, 0, −0.2       48 zero’s 24 one’s

u(1, 3) =

.

(9.57)

(9.58)

The diagonal matrix .D e is   D e = diag h∗p1 , h∗p2 , h∗p3 , 3000, 1500 ∈ R74×74

.

The resulting linear system reads ⎡

⎤⎡

⎢ ⎢ ⎣

⎥⎢ ⎥⎢ ⎦⎣

−I74×74 Ae74×2

.

e D74×74

074×2

e π74×1

μe2×1

⎤

⎤ −Ψ (h∗ )72×1 ∗ ⎥ ⎥ ⎢ ⎥ = ⎢ Θ[Q ]2×1 ⎥ ⎦ ⎦ ⎣ ⎡

(9.59)

074×1

where .Θ[Q∗ ] = (40, 25)T . The resulting system of linear equations is easily solved. The dual variable e .π associated with inequality constraints is displayed graphically in Fig. 9.10. By observing that .π e ≥ 0, we are assured the discrete solution .(h∗ , Q∗ ) of the variational inequality is in fact an elastic demand dynamic user equilibrium.

370

9 Numerical Results

100 90 80

Value

70 60 50 40 30 20 10 0

0

10

20

30

40

50

60

70

Index Fig. 9.10 SRDT DUE with elastic demand: the dual variable .π e associated with inequality constraints in the variational inequality. The horizontal axis indicates the order at which each element appears in the vector .π e ; the vertical axis indicates the value of the corresponding element

9.2 Basic Setup of Numerical Examples of Dynamic User Equilibria In the following sections, we present some computational results for the simultaneous route-and-departure-time (SRDT) choice dynamic user equilibria on networks of varying sizes and shapes. Section 9.3 focuses on DUEs with fixed travel demand, and 9.4 and 9.5 present computational results of DUEs with elastic demand and bounded rationality, respectively. As our computational algorithms are path-based, explicit path generation is required for each origin-destination pair. Following Friesz et al. (1992) and Friesz et al. (1993), we apply the Frank-Wolfe algorithm to generate the path set, by making the network increasingly congested through uniformly scaling up the O-D demand matrix and saving path information produced by the F-W algorithm. This method is intrinsically similar to column generation as it only calculates paths as they are used. Route sets generated in this way encapsulate essential network information such as O-D demand tables, network topology, and interactions of O-D flows through congestion; they are arguably more reasonable than others such as those produced by k-shortest-path algorithms, which only consider free flow link travel times with no consideration of the distribution of O-D demands or their interactions on the network. For all subsequent computational experiments, the dynamic network loading submodule is based on the DAE system (see Sect. 8.2.4). The DAE system is

9.3 Numerical Solutions of DUE with Fixed Demand

371

derived from the variational solution representation of the LWR model, as well as its network extension. Not only is such a DNL model capable of capturing realistic network traffic dynamics, including shock waves and vehicle spillback, it also facilitates computations on large-scale networks due to the simplified solution representation and streamlined numerical procedures. More details of the DNL model and open-source codes are provided in Han et al. (2019).

9.3 Numerical Solutions of DUE with Fixed Demand We employ the fixed-point algorithm (Sect. 6.1) to illustrate that this method tends to exhibit satisfactory empirical convergence with a limited number of iterations, despite the fact its proof of convergence requires strong monotonicity. Four test networks are illustrated in Fig. 9.11 and described in Table 9.2. In particular, the Nguyen network was initially studied in Nguyen (1984). The other three networks are based on real-world cities in the USA, although different levels of network aggregation and simplifications have been applied. Detailed network parameters, including coordinates of nodes and link attributes, are sourced from Bar-Gera (2016).

9.3.1 Performance of the Fixed-Point Algorithm The termination criterion for the fixed-point algorithm is % k+1 %2 %h − hk % ≤ . % %2 %hk %

(9.60)

where .hk denotes the path departure vector in the k-th iteration. The threshold . is set to be .10−4 for the Nguyen and Sioux Falls networks and .10−3 for the Anaheim and Chicago Sketch networks. These different thresholds were chosen to accommodate the varying convergence performances of the algorithm on different networks (see also Fig. 9.12). Table 9.3 summarizes the computational performance of the fixed-point algorithm and DNL procedure on different networks based on the termination criterion (9.60). It is shown that the number of fixed-point iterations is not significantly impacted by the disparate sizes of the test networks, which suggests the scalability of the algorithms. Figure 9.12 shows the relative gaps, i.e., left-hand side of (9.60), for a total of 100 fixed-point iterations on the four networks. It can be seen that for relatively small networks (Nguyen and Sioux Falls), convergence can be achieved relatively quickly

372

9 Numerical Results 3

1

2

1 2

1

1

2

12

8

3

4

3

4

5

7

11

4

9

6 10

35

15

5 13

23 9

31

12 21

5

6

10

6

7

11

8

7

33

12

8 13

12

27

11

36 34

32

40

16 19 8

24

16 29 5149 52 30 17 28 43

9

7

20

48

10

17

47

22

25 26

4

14 6

18 54 55

18

50

53 58

9

14

10

15

11

16

2

37

42

18

17

41

14

38

44 71

19

3

73 13

Nguyen network (13 nodes, 19 links, 4 zones)

Anaheim network (416 nodes, 914 links, 38 zones)

74 39

76 24

19

45

56

60

46 67 72 70

23

13

57

15

69 66 75

59 61

22 65 21

63 68 62 64

20

Sioux Falls network (24 nodes, 76 links, 24 zones)

Chicago sketch network (933 nodes, 2950 links, 387 zones)

Fig. 9.11 The four test networks for DUE algorithms

and to a satisfactory degree; also the corresponding curves are monotonically decreasing and smooth. For the Anaheim and Chicago Sketch networks, the decreasing trend of the gap can sometimes stall and experience fluctuations locally.

9.3 Numerical Solutions of DUE with Fixed Demand

373

Table 9.2 Key attributes of the test networks No. of links No. of nodes No. of zones No. of O-D pairs No. of paths 1

Nguyen network 19 13 4 4 24

Sioux Falls 76 24 24 552 6180

Anaheim 914 416 38 1406 30, 719

10

Nguyen network

Chicago Sketch 2950 933 387 86, 179 250, 000

Sioux Falls network

1

1E-1

Relative Gap

Relative Gap

1E-1

1E-2

1E-3

1E-2

1E-3

1E-4

1E-4 0

20

40 60 Iteration Number

80

1E-5

100

20

Anaheim network

1

40 60 Iteration Number

80

100

80

100

Chicago Sketch network 1

Relative Gap

1E-1

Relative Gap

0

1E-2

1E-1

1E-2

1E-3 1E-3

1E-4 0

20

40 60 Iteration Number

80

100

0

20

40 60 Iteration Number

Fig. 9.12 The relative gaps (base 10 logarithm) within 100 fixed-point iterations. Here, 1E-x means .10−x Table 9.3 Performance of the fixed-point algorithm on different networks No. of iterations Computational time Avg. time per DNL Avg. time per FP update

Nguyen network 54 5.9 s 0.1 s 0.007 s

Sioux Falls 73 6.1 min 4.3 s 0.6 s

Anaheim 45 23.3 min 25.7 s 3.1 s

Chicago Sketch 69 4.8 hr 163.9 s 81.2 s

9.3.2 DUE Solutions In this section we examine the DUE solutions obtained upon convergence of the fixed-point algorithms. We begin by selecting four arbitrary paths per network to illustrate the properties of the solutions. Figure 9.13 shows the path departure rates

9 Numerical Results

3

16

0 4

600

12

400

8

200

4

0

5

1

2

3

Path 2672

4

10

5

20

0

1

2

3

2

3

4

5

0

0

5

0

1

2

3

10

40

0 1

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374

0 0

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3

4

5

Time (hr)

(Chicago Sketch network)

Fig. 9.13 Path departure rates and corresponding effective path delays (travel costs) of selected paths in the DUE solutions on the four test networks

as well as the corresponding effective path delays. We observe that the departure rates are non-zero only when the corresponding effective delays are equal and minimal, which conforms to the notion of DUE. Note that the bottoms of the effective delay curves should theoretically be flat, indicating equal travel costs. This is not exactly the case in the figures since we can only obtain approximate DUE solutions in a numerical sense, given the finite number of fixed-point iterations performed to reach those solutions. To rigorously assess the quality of the DUE solutions, we define the gap function between each O-D pair .(i, j ) ∈ W as   . GAPij = max Ψp (t, h∗ ) : t ∈ [t0 , tf ], p ∈ Pij such that h∗p (t) > 0   − min Ψp (t, h∗ ) : t ∈ [t0 , tf ], p ∈ Pij such that h∗p (t) > 0

.

(9.61) Here, .GAPij indicates the range of travel costs experienced by travelers between O-D pair .(i, j ). In an exact DUE solution, the gap should be zero for all O-D pairs. Figure 9.14 summarizes all the O-D gaps of the DUE solutions on the four test networks. It can be seen that the majority of the O-D gaps are within 0.2

9.3 Numerical Solutions of DUE with Fixed Demand 2

375

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Count

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O-D Gap

(Chicago Sketch network)

Fig. 9.14 Distributions of O-D gaps corresponding to the DUE solutions. The O-D gap is calculated according to (9.61)

across all four networks. Even for large-scale networks (Anaheim and Chicago Sketch), the 75th percentiles are within 0.16, and the whiskers extend to 0.3. A comparison between Anaheim and Chicago Sketch also reveals that the latter yields better solution quality in terms of O-D gaps, despite the obviously larger size of the problem. This suggests that solution quality is not necessarily compromised by the size of the network.

376

9 Numerical Results

9.4 Numerical Examples of DUE with Elastic Demand This section demonstrates solutions of the DUE problems with elastic demand (E-DUE). We also compare three algorithms for computing E-DUEs: fixed-point algorithm (Sect. 6.1.2), self-adaptive projection algorithm (Sect. 6.2.2), and proximal point method (Sect. 6.3.2). All the computations reported here were performed within MATLAB on a standard laptop with 4 GB of RAM.

9.4.1 The Seven-Link Network The first test network has seven links and three paths and is shown in Fig. 9.15. We assume the following inverse demand function v=−

.

Q + 1.2 2000

for O-D pair .(1, 6). The following termination criteria are employed for the three computational algorithms. For the projection method and the proximal point method, the algorithm is terminated if the relative gap obeys .

Xk+1 − Xk E ≤ 10−5 Xk E

(9.62)

. where .Xk = (hk , Qk ). For the self-adaptive projection method, the termination criterion is .

r(Xk ; βk )E ≤ 10−6 Xk E

(9.63)

where the residual .r(Xk ; βk ) is given in (6.46). We apply the projection method (PM), the self-adaptive projection method (SA), and the proximal point method (PP) to solve the E-DUE problem on the seven-link network shown in Fig. 9.15. Under the termination criteria (9.62) and (9.63), the algorithms converge after 200 (PM), 3903 (SA), and 180 (PP) iterations. We use Fig. 9.15 The seven-link network

3 I2 1

I5

I1

I7

I4

2 I3

4

5 I6

6

9.4 Numerical Examples of DUE with Elastic Demand

377 1E−1

1E−2

1E−2

Relative gap

1E−4

Proximal Point Mehtod

Self−adaptive Projection

1E−4

1E−3

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Relative gap

Projection Method

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1E−3 1E−4

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1E−5

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50

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150

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1000

Iteration counter

2000

3000

1E−6 0

4000

50

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150

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Iteration counter

Fig. 9.16 The seven-link network: relative gaps, defined in (9.62) and (9.63), at each iteration of the algorithms. Here, 1E-x means .10−x SA

1000 800 0

50

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k

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(Q , ν )

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ν=−Q/2000+1.2

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1

1000 2000 3000 Iteration counter

1.5

k

(Q , ν )

PM

0.5 0 0

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PP

1

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1

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SA

PM Travel cost

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1200

1.5

1.5

Travel cost

PP

1200

Demand

Demand

Demand

PM 1200

0 800

1400

1400

1400

ν=−Q/2000+1.2

1 0.5 0 800

1000

1200 Demand

1400

1600

Fig. 9.17 The seven-link network: the elastic demand .Qk and the average travel cost .v k at each iteration. The initial value for the demand is always set to be .Q0 = 1000. PM, projection method; SA, self-adaptive projection; PP, proximal point

Fig. 9.16 to show the left-hand sides of (9.62) and (9.63) at each iteration of the algorithms. It can be seen that although all three algorithms reach the termination threshold within a finite number of iterations, their convergence speeds vary. In particular, the self-adaptive projection method has much slower yet smoother convergence than the other two; and its relative gap is monotonically decreasing as the iterative process continues. The projection method and the proximal point method show qualitatively similar convergence trends, although their relative gaps decrease much more quickly than the self-adaptive method, but may have some local increase and oscillation. We now turn to the solutions produced by these algorithms. Figure 9.17 illustrates two main quantities: the elastic demand .Qk generated at each iteration and the

378

9 Numerical Results

drivers’ average travel cost at each iteration defined as

1 tf Ψp (t, hk ) · hkp (t) dt Qk t0

. vk =

.

k = 0, 1, 2, . . .

p∈P

The reason for using the average travel cost .v k is as follows. Until an E-DUE has been found (as allowed by the prescribed tolerance), the travel costs within the same O-D pair are not equilibrated; thus we use the averaged travel cost to demonstrate the demand-cost relationship stipulated by the inverse demand function. It can be seen from Fig. 9.17 that both .Qk and .v k converge to a fixed value when the algorithms terminate, and they converge to a state that is consistent with the inverse demand function (see the bottom row of Fig. 9.17). It remains to show that the experienced travel costs within the same O-D pair for all route and departure time choices are equal and minimized. This is illustrated in Fig. 9.18, where we show the path departure rates and the corresponding travel costs for all three paths in the network. We see that the travel costs are indeed equal and minimized whenever the departure rates are positive, which confirm equilibrium has been reached.

0 0

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Flow (veh/h)

1000

Path travel cost

Fig. 9.18 The seven-link network: path departure rates and associated travel costs for all three paths. PM, projection method; SA, self-adaptive projection; PP, proximal point

9.4 Numerical Examples of DUE with Elastic Demand

379

9.4.2 The Sioux Falls Network We further test the three algorithms on the Sioux Falls network (see Fig. 9.11) with 6 O-D pairs and 119 paths. Similar to the seven-link network, we assume the following linear inverse demand function for all the six O-D pairs: v=−

.

Q + 1.6 500

We start with the algorithm convergence following the same criteria (9.62) and (9.63). Figure 9.19 shows the relative gap at each iteration of the algorithm. The number of iterations needed for the three algorithms is 469 (PM), 2304 (SA), and 109 (PP). Observations regarding the convergence trends are similar to the previous test network: the self-adaptive method has a slower yet smoother convergence rate, and the relative gap is monotonically decreasing. The projection method and the proximal point method have a faster convergence, but their relative gaps may have some local oscillations. The convergence of the elastic travel demands .Qkij and the O-D-specific average travel costs .vijk is demonstrated in Fig. 9.20, which has a similar style of presentation as Fig. 9.17 but shows information for all six O-D pairs. We can, again, confirm that .Qkij and .vijk converge to the relationship stipulated by the inverse demand function for all O-D pairs when the algorithms converge. However, there are some inaccuracies in the convergence of the self-adaptive method and the proximal point method as the trajectories of .(Qkij , vijk ) did not reach the line .v = −Q/500 + 1.6 when the algorithms terminated. This may, again, be overcome by tighter termination tolerances. 1E−3

Self−adaptive Projection

Projection Method

Proximal Point Method

1E−5

1E−3

1E−4

Relative gap

Relative gap

Relative gap

1E−2

1E−6

1E−5

0

200

400

Iteration counter

600

1E−4

1E−5

0

1000

2000

Iteration counter

3000

0

50

100

150

Iteration counter

Fig. 9.19 The Sioux Falls network: relative gaps, defined in (9.62) and (9.63), at each iteration of the algorithms. Here, 1E-x means .10−x

380

9 Numerical Results

1000

1000

1000

Travel cost

Travel cost Travel cost

1.5 1 0.5 0 400

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OD1

Demand

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OD3

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1000

ν = −Q/500+1.6

Fig. 9.20 The Sioux Falls network: The elastic demands .Qkij and the average travel costs .vijk for each O-D pair .(i, j ) at each iteration. The initial values for the demands are always set to be 0 .Qij = 1000 for all .(i, j ) ∈ W . PM, projection method; SA, self-adaptive projection; PP, proximal point Table 9.4 Summary of the three algorithms for computing E-DUE

Projection method Self-adaptive Proximal point

Seven-link Iteration # 200 3903 180

DNL # 200 7805 5400

Time 3s 108 s 74 s

Sioux Falls Iteration # 469 2304 109

DNL # 469 4607 2332

Time 382 s 3810 s 2425 s

9.4.3 Algorithm Performance We summarize the performances of the three algorithms in terms of solution quality, convergence rate, computational complexity, and computational time. Table 9.4 shows such basic information on the computational performances of each. In general, the self-adaptive projection method takes much more iterations to reach a prescribed convergence threshold than the other two methods. However, one should take into account the fact that the self-adaptive method utilizes a different termination criterion than the other two; i.e., it relies on the residual function

9.5 Numerical Examples of DUE with Bounded Rationality

381

r(Xk ; β k ), while the others focus on the distance between two consecutive iterates (see (9.62) and (9.63)). The self-adaptive method and the proximal point method require multiple dynamic network loading (DNL) procedures to be performed within a single iteration, and thus end up with more DNL problems solved than the projection method. In our computation, the average numbers of DNLs performed within one iteration are roughly 2 (self-adaptive) and 26 (proximal point). The proximal point method requires solving a regularized VI within each iteration and thus demands substantially more computational effort. The projection method requires the least computational effort and yields good solution quality despite its restrictive convergence condition (strong monotonicity). The self-adaptive method and the proximal point method require more iterations and DNL procedures to converge, which highlights a trade-off between theoretical convergence and computational complexity. Overall, all three algorithms have successfully computed E-DUE solutions with convergence according to a prescribed stopping threshold, despite the fact that their convergence conditions have not been verified for the network performance model.

.

9.5 Numerical Examples of DUE with Bounded Rationality This section demonstrates solutions of DUE problems with bounded rationality (BR-DUE). Similar to Sect. 9.4, we compare the three algorithms for computing E-DUEs: fixed-point algorithm (Sect. 6.1.3), self-adaptive projection algorithm (Sect. 6.2.3), and proximal point method (Sect. 6.3.3). All the computations reported here were performed within MATLAB on a standard laptop with 4 GB of RAM.

9.5.1 VT-BR-DUE on a Seven-Link Network Following the variable tolerance BR-DUE model presented in Sect. 3.4.2, the first computational example is meant to illustrate its solution where the user tolerances are endogenous, depending on the realized departure flows. The test network shown in Fig. 9.21 has one O-D pair .(1, 6) and three paths .p1 {a, c, e, g}, .p2 = {a, b, d, e, g}, .p3 = {a, b, f, g}. The fixed travel demand is 2000 (in vehicles). We consider two cases wherein the following two sets of tolerance functions are considered:   100 [Case I] .ε1 = ε3 ≡ 0.1, .ε2 = 0.15 1 − 100+V 2   100 [Case II] .ε1 = ε3 ≡ 0.1, .ε2 = 0.2 1 − 100+V 2

382

9 Numerical Results

f a

1

2

b

d

3

4

e

g 5

6

c Fig. 9.21 The seven-link, six-node test network for VT-BR-DUE

ε2

ε2 0.2 0.15

0

V2

Case I

0

V2

Case II

Fig. 9.22 The VT-BR-DUE problem: the functional forms selected for .ε2 (·)

where &.εi denotes the user tolerance associated with path .pi , .i = 1, 2, 3, and . t V2 = t0f hp2 (t) dt is the total traffic volume on path .p2 . In both cases, the cost tolerance .ε2 is a functional of .hp2 (·); in fact, it is an increasing function of .V2 . In addition, .ε2 (·) is bounded from above by a fixed constant; see Fig. 9.22 for these functional forms. Notice that Case II yields a higher tolerance along .p2 than Case I, provided the same value of .V2 is used. By comparing the two cases, we will show how the slight difference in the tolerance functions manifests itself in the solution. We further note that these chosen functional forms for the tolerances are for illustration purposes only, while further study is clearly needed to formulate and calibrate those specific functional forms needed to realistically capture routing behavior, which is beyond the scope of this book. The VT-BR-DUE problems were solved with the fixed-point algorithm (Sect. 6.1.3), and the results are displayed in Fig. 9.23, where we show the departure rates along the three paths and the corresponding effective path delays. In these numerical solutions, the traffic volumes on path .p2 are, respectively, .V2 = 1105 (veh) in Case I and .V2 = 1178 (veh) in Case II. Accordingly, the corresponding tolerances are:

.

[Case I] .ε1 = ε3 = 0.1, .ε2 = 0.1376 [Case II] .ε1 = ε3 = 0.1, .ε2 = 0.1844 The minimum effective delay between the O-D pair, denoted .v16 , and the tolerance thresholds .v16 + εi , .i = 1, 2, 3, are shown in Fig. 9.23. From this figure we see that

2000

Departure Rates (veh/h)

Departure Rates (veh/h)

9.5 Numerical Examples of DUE with Bounded Rationality

Path 1 Path 2 Path 3

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5

Fig. 9.23 Solutions of the VT-BR-DUE problems in Case I (left) and Case II (right). In the second row, the solid horizontal line represents the minimum effective delay .v16 , the dashed lines represents .v16 + ε2 , and the dotted lines represent .v16 + ε1 (or .v16 + ε3 )

0.02 Case I Case II Relative Gap

0.015

0.01

0.005

0

0

20

40

60

80 100 120 140 Iteration Counter

160

180

200

Fig. 9.24 Convergence of the fixed-point algorithm for the seven-link, six-node network

the computed solutions are indeed solutions of the VT-BR-DUE problems since h∗pi (t) > 0 implies that .Ψ (t, h∗pi ) ≤ v16 + εi , .i = 1, 2, 3. We also observe that as a result of the chosen forms of .ε2 (·), the total traffic volume on path .p2 is smaller in Case I than in Case II. The reason is that, if given the same traffic volume .V2 , drivers following path .p2 have a lower tolerance in Case I than in Case II, and thus more drivers are likely to switch to the other paths in Case I. Finally, for both computational scenarios, the fixed-point algorithm converged after a finite number%of iterations.% This ' %can%be seen from Fig. 9.24, where the relative gap is expressed by .%hk+1 − hk %L2 %hk %L2 .

.

384

9 Numerical Results

9.5.2 BR-DUE on the Nguyen Network The second example analyzes the impact of the fixed cost tolerance .ε on the final solution of the BR-DUE problem. The numerical results demonstrate the sensitivity of the BR-DUE solution to the choices of .ε; they also highlight the importance of a well-calibrated indifference band when BR-DUE models are applied. The second test network is the Nguyen network shown in Fig. 9.11, with 4 origindestination pairs and 24 paths. We apply a fixed tolerance .ε for all the O-D pairs and all the paths, so that the problem becomes a BR-DUE problem. Two values of .ε are considered: (1) .ε = 0.4 and (2) .ε = 0.2. The corresponding BR-DUE solutions are partially illustrated in Fig. 9.25 for .ε = 0.4 and in Fig. 9.26 for .ε = 0.2. Notice that the revised effective delays in all these figures refer to the function .φpε (·, h∗ ) defined in (3.52). Recall that a BR-DUE solution .h∗ must solve the following variational inequality ( .

) φ ε (·, h∗ ), h(·) − h∗ (·) ≥ 0

∀h ∈ Λ

500

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According to our earlier discussion following Corollary 3.1, we are assured that the solutions presented in Figs. 9.25 and 9.26 are indeed BR-DUE solutions, since ∗ ε ∗ .hp (t) > 0 implies that .φp (t, h ) is equal and minimal. We also see from a comparison between Figs. 9.25 and 9.26 that the BR-DUE solutions are very different as a result of different values of .ε, indicating a high sensitivity of the solution to the tolerances. Therefore, it is crucial to identify appropriate values of the cost tolerance in order to accurately predict and describe

Time (hour)

Fig. 9.25 The 19-link network with .ε = 0.4: BR-DUE path departure rates and the corresponding revised effective path delay .φ ε , which is defined in (3.52)

3000

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9.5 Numerical Examples of DUE with Bounded Rationality

Time (hour)

Fig. 9.26 The 19-link network with .ε = 0.2: BR-DUE path departure rates and the corresponding revised effective path delay .φ ε , which is defined in (3.52)

the path departure rates. In addition, as we shall demonstrate in Sect. 9.5.3, the larger the value of .ε, the fewer iterations the fixed-point algorithm usually takes to converge.

9.5.3 BR-DUE on the Sioux Falls Network The third test network is the Sioux Falls network illustrated in Fig. 9.11. In this network we consider 6 origin-destination pairs, among which 119 paths are selected. The main purpose of this numerical study is to evaluate and compare the performances of the three computational algorithms (fixed point, self-adaptive projection, proximal point), in terms of their convergence and computational efficiency. To make our numerical study more comprehensive, we also consider the seven-link network and the Nguyen network. Moreover, we will compare convergence results with varying values of .ε. To simplify the problem, we assume that the cost tolerances are fixed and equal for all O-D pairs and paths. The following termination criteria are employed for the three methods: .

hk+1 − hk L2 ≤ 10−4 (fixed-point method; proximal point method). hk L2 % % k %r(h ; βk )% 2 L ≤ 10−4 (self-adaptive projection method) hk L2

(9.64) (9.65)

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400

150

Fixed−point method Time (s)

Iteration

Fixed−point method 200

0

0

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0.2

0.3

100 50 0

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Self−adaptive method

Time (s)

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400

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0.2

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Proximal point method

Proximal point method

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0.4

500

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60 40 20

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Seven−link network

0.1

0.2

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Tolerance 19−link network

Sioux Falls network

Fig. 9.27 Comparison of the three algorithms on the three test networks with a range of .ε

The convergence results and the computational times of the three algorithms implemented on different networks are shown in Fig. 9.27, where we consider a range of values for the tolerance .ε. The three algorithms show qualitatively different convergence trends. For the fixed-point algorithm and the proximal point method, as the tolerance .ε becomes larger, both algorithms take fewer iterations to converge, although the iterations needed by the proximal point method are less sensitive to the tolerance than the fixed-point method. This is understandable: given that the larger the tolerance, the more likely the traffic is equilibrated. For the self-adaptive projection method, however, there is no discernible effect of .ε on the convergence of the algorithm, especially for the smaller networks. We also see, from the first and the third rows of Fig. 9.27, that as the network size increases, despite the increase in the dimension of the problem (i.e., the number of paths), the number of iterations required by both algorithms remains roughly the same. This implies good scalability and the dimension-free nature of the fixed-point and proximal point methods. With regard to the computational time, the three algorithms differ significantly. In particular, for the same network, the computational time of the fixed-point method is proportional to the number of iterations needed. This is obviously due to the fact that each fixed-point iteration requires exactly one DNL procedure. When the computational times of the fixed-point algorithm are compared across different networks, the larger the network, the more time needed even if the iteration numbers remain more or less the same. This is because the DNL procedure for larger

9.5 Numerical Examples of DUE with Bounded Rationality

387

networks consumes more time. For the self-adaptive method and the proximal point method, the computational times are disproportional to the iteration numbers; this is because each iteration in these algorithms may require multiple DNL procedures (see Step 2 of the self-adaptive projection method and Step 1 of the proximal point method). As a result, their computational times are significantly larger than the fixed-point algorithm, especially for the largest network (Sioux Falls). We may conclude that the self-adaptive method and the proximal point method in general take more time to reach a given level of convergence than the fixed-point algorithm, despite the fact that they enjoy more relaxed convergence conditions than the latter. This highlights the potential trade-off between improved convergence and computational efficiency. To further analyze the convergence patterns of these three methods, we have shown in Fig. 9.28 the relative gaps defined in (9.64) and (9.65) at each iteration. This figure is based on the calculation on the Sioux Falls network with tolerance .ε = 0.2. We see that the self-adaptive projection and the proximal point method have smoother convergence than the fixed-point method. The latter, however, has a faster convergence rate. Overall, the fixed-point method and the proximal point method converge faster than the self-adaptive projection method within finite iterations, and they are able to achieve a relative gap of .10−6 within 300 iterations. However, distinctions need to be made between the asymptotic convergence of an algorithm and the convergence to a given precision within finite iterations. Most existing convergence results are established in the former sense, while only a few (such as Theorem 6.11) are concerned with convergence to an approximate solution in a computationally practical way.

E+1 Fixed−point method Self−adaptive projection Proximal point method

1

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E−1 E−2 E−3 E−4 E−5 E−6 E−7

0

50

100

150

200

250

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Iteration Counter Fig. 9.28 Convergence of the three algorithms on the Sioux Falls network with tolerance .ε = 0.2. On the y-axis, E-x means .10−x

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References and Suggested Reading Bar-Gera, H. (2016). Transportation network test problems. https://github.com/bstabler/ TransportationNetworks. Friesz, T. L., Cho, H., Mehta, N., & Tobin, R. (1992). Simulated annealing methods for network design problems with variational inequality constraints. Transportation Science, 26(1), 18–26. Friesz, T. L., Tobin, R., Shah, S., Mehta, N., & Anandalingam, G. (1993). The multiobjective equilibrium network design problem revisited: A simulated annealing approach. European Journal of Operational Research, 65, 44–57. Han, K., Eve, G., & Friesz, T. L. (2019). Computing dynamic user equilibria on large-scale networks with software implementation. Networks and Spatial Economics, 19(3), 869–902. Nguyen, S. (1984). Estimating origin-destination matrices from observed flows. In Florian, M. (Ed.), Transportation Planning Models (pp. 363–380). Amsterdam: Elsevier Science Publishers. Yperman, I., Logghe, S., & Immers, L. (2005). The link transmission model: An efficient implementation of the kinematic wave theory in traffic networks, advanced OR and AI methods in transportation. In Proceedings of the 10th EWGT meeting and 16th Mini-EURO conference (pp. 122–127). Publishing House of Poznan University of Technology: Poznan, Poland.

Index

A Algorithm, vi, 2, 4, 6, 7, 14, 15, 22, 23, 26, 101, 107, 125, 129, 195, 199–241, 243, 252, 284, 303, 370–373, 376–383, 385–387

B Boundedly rational dynamic user equilibrium, 106–109, 177, 183, 207, 216 Bounded rationality (BR), v, 26, 107–109, 125, 139–147, 177–197 Bounded variation (BV), 303, 339, 345

C Calculus of variations, 33, 45–48, 84, 125 Characteristics, 1, 76–81, 108, 176, 314, 315 Compactness, 36–38, 160, 162–164, 170, 183, 189 Conservation law, 20, 33, 73–86, 265–267, 295–297, 303–306, 315, 329, 332 Continuity, 51, 52, 158, 161, 164, 179, 182, 183, 185, 188, 189, 193, 196, 199, 213, 215, 218, 225–228, 230, 231, 240, 252–259, 271, 275, 276, 284–288, 329–348 Continuous time, 4, 7, 14, 22–25, 37, 48–50, 53–58, 91, 132, 155, 186,188, 261, 262, 272, 278, 318, 319, 324, 325, 327–328, 347 Contraction mapping theorem, 38 Convergence, vi, 3, 7, 14, 15, 21, 22, 26, 101, 164, 166, 167, 172, 173, 199–203, 207, 208, 210, 211, 213,

215, 218–241, 243, 253, 255–259, 284–288, 371, 373, 377, 379–381, 383, 385–387 Cumulative departures, 21, 126, 132, 174, 175

D Delay operator, 4, 7–9, 14, 20, 21, 23, 26, 91, 93, 95, 101, 108, 157–159, 161, 178, 179, 182, 186, 187, 192, 196, 211, 225, 226, 232, 241, 243, 252–259, 284–288, 329–348 Demand, 95–100, 115–118, 125–131, 157– 167, 199–202, 211–213, 218–220, 223, 226, 299–300, 314–317, 353, 355–362, 371–375 Departure rates, v, 1, 4, 6–9, 17, 24, 91–94, 96, 103, 108–110, 113, 114, 126, 127, 132, 133, 140, 141, 144, 148, 155–159, 161, 166, 169, 171, 172, 174–177, 186, 187, 202, 206, 207, 210, 212, 215–217, 220, 222–224, 229, 249, 251, 259, 284, 287, 288, 304, 308, 310, 317, 340, 344–347, 355, 373, 374, 378, 384, 385 Destination, 13, 26, 94, 140, 156, 158, 159, 176, 306–309, 339, 340, 342,347 Differential algebraic equation (DAE), 4–7, 22, 250, 251, 281–283, 311,313–320, 328, 370 Differential variational inequality (DVI), v, 9, 15, 22, 26, 61–66, 70, 72,101, 125–152, 155, 169, 199 Discrete time, 115–117, 119–121, 187

© Springer Nature Switzerland AG 2022 T. L. Friesz, K. Han, Dynamic Network User Equilibrium, Complex Networks and Dynamic Systems 5, https://doi.org/10.1007/978-3-031-25564-9

389

390 Diverge junction, 300–301, 322, 324–325, 327, 329, 330, 332–335, 340,342, 344, 345, 347 Dynamic network loading, v, 4, 6, 9–26, 86, 91, 156, 158, 161, 196, 202, 206, 210, 212, 213, 215–218, 220, 222, 224, 226, 243–290, 295–348, 370,381 Dynamic user equilibrium (DUE), v, vi, 1–3, 6–8, 10, 12–15, 18, 21–26, 33, 37, 41, 91–122, 125–153, 155–197, 199–241, 248, 252, 261, 353–387

E Effective path delay, 7, 14, 16, 21, 24, 93, 105, 108–110, 114, 115, 120, 140, 143, 144, 147, 159–161, 164, 169, 179, 180, 186, 190, 192, 202, 206, 210, 220, 222, 224, 236, 252, 259, 285, 288, 358, 361, 364, 367, 369, 374,382, 384, 385 Elastic demand, v, 101–106, 118–122, 125, 131–139, 167–173, 202–207, 213–216, 220–222, 225, 226, 232, 353, 362–370, 375–381 Endogenous tolerance, 107, 110–115, 139, 144–147, 151, 177, 223, 381 Equilibrium, 1, 67, 69, 96, 100, 109, 133, 140, 155, 230, 335, 355, 356, 358, 359, 361, 368, 378 Existence, 3, 23, 25, 26, 37, 51, 61, 62, 79, 101, 106, 108, 110, 155–197, 206, 243, 252, 259, 283, 284, 288, 303 Exogenous tolerance, 109–115, 140–143, 148–150, 177, 207, 216

F Fixed-point, vi, 4, 22, 37, 38, 67, 101, 125, 129–131, 136–139, 148–153, 155, 158, 169, 199–210, 213, 219, 221, 224–232, 240, 371–374, 376, 381–383, 385–387 Fritz John conditions, 40 Fundamental diagram (FD), 9, 21, 74–76, 86–88, 173, 175, 265, 295, 296, 305, 312–321, 325, 330, 344, 345

G Generalized Lax-Hopf formula, 84–89, 311 Generalized Nash equilibrium, 67, 68 Generalized Vickrey model, 276–281, 283, 284

Index H Hamilton-Jacobi equation, 83–86, 266, 267, 311, 312 Hilbert space, 33–35, 37, 91, 92, 97, 101, 102, 106, 126, 136, 157, 167–169, 197, 199, 202, 211, 220, 231, 259

I Ill-posedness, 330–332

J Junction model, 243, 296, 301, 306, 314, 316, 322–325, 329–339, 344

K Kuhn-Tucker conditions, 40–45, 115–122

L Lax-Hopf formula, 22, 84–85, 88, 89, 266, 267, 314, 315 Linear complementarity, 11, 261–262 Link transmission model (LTM), 23, 295, 319, 325–328

M Merge junction, 300–303, 323–324, 340–343, 347 Minimum norm projection, 35, 130, 138, 148, 151, 199, 200, 203, 208, 227,230, 232, 324 Monotonic, 7, 23

N Nash equilibrium, 66–70, 72, 73, 174–176 Nguyen network, 371, 373, 384–385 Nonlinear program, 25, 39–45 Non-physical queue, 243–290

O Optimal control, vi, 2, 13, 24, 26, 33, 48–62, 64, 66, 69, 73, 125–128, 130, 133, 137, 138, 141, 145, 149, 151, 152, 200, 204, 208 Ordinary node, 305–307 Origin, v, 14, 251, 302, 306–310, 312–313, 345–347

Index P Partial differential algebraic equation (PDAE), 5, 6, 17–20, 304–312, 317, 330, 340, 344 Path delay, 4, 7, 8, 15, 17, 23, 91, 93, 95, 101, 129, 157, 159, 186, 225, 226, 232, 236, 244, 249, 252, 283, 285, 287, 288, 304, 329, 347–348, 357 Path disaggregation variables (PDVs), 20, 305, 307–310, 335, 338, 339, 344–345, 347 Path flows, 5, 7, 9, 17, 37, 92, 94–96, 102, 103, 108, 109, 116, 119, 126, 140, 141, 144, 155, 156, 196, 197, 201, 209, 243, 251, 252, 284, 308, 355, 356, 358, 359, 361–363, 368 Path travel time, 93, 157, 309, 310, 317, 329, 347 Physical queue, 3, 243, 295–348 Projection, 34, 35, 136, 138, 148, 151, 199, 200, 203, 208, 211–214, 216–218, 225, 226, 230, 232, 233, 236, 302, 324, 376–381 Proximal point, 125, 218–226, 238–241, 376–381, 385–387 Q Queuing model, 345–346 R Riemann solver, 17, 296, 298–304, 306, 307, 313, 314, 316, 322, 324, 325, 330 S Self-adaptive projection, 125, 211–218, 226, 233–238, 240, 376–378, 380, 381, 385, 387 Shock wave, 23, 76, 78, 79, 81, 176, 295, 314, 315, 371 Simultaneous route and departure time (SRDT), 6, 7, 24–26, 91, 97, 100, 103, 109, 116, 120, 126, 129, 155–158, 177, 183, 284, 355–360, 362–368, 370 Sioux Falls network, 22, 371, 379–380, 385–387

391 Square integrable functions, space of, 35, 51, 62, 92 Strictly monotonic, 225, 255 Strong continuity, 164, 254, 259, 288 Strongly monotonic, 199, 221, 225–227, 229, 371, 381 Sufficiency, 41, 57–61, 98, 105, 113, 128, 133, 134, 141, 145, 149, 152, 232 Supply, 3, 22, 263, 296, 299–301, 308, 312, 314–319, 323, 326, 327, 329, 331, 339–344, 347

T Tolerance, 107–110, 113, 140, 144, 147, 148, 150, 177–179, 182, 183, 191, 192, 207, 209, 216, 219, 221, 223, 239, 378, 379, 381–387 Topological vector space, 35–36, 158, 162

V Variation, 46–48, 53–57, 304, 335, 338, 344, 345, 347, 348 Variational formulation, 174, 311–320, 327 Variational inequality, 2, 4, 7, 13, 21, 22, 25, 41, 43–45, 51, 57, 62, 63, 67, 91–122, 125, 132, 155, 167–169, 173, 177, 187, 211, 213, 216, 218, 219, 221, 223, 226, 230, 248, 362, 367, 369, 370 Variational principle, 22, 23, 83–89, 311, 327 Variational theory, 23, 33, 84–89, 311–312 Vickrey model, 10–12, 160, 179, 259–288, 308, 353, 354 W Wardrop, J.G., 1, 23, 106–108 Wave front tracking (WFT), 299, 303, 304, 329, 333, 335, 336 Weakly monotonic, 226–232 Weak solution, 74, 76–81, 297, 303, 331 Well-posedness, 329–339, 345–347 Within-link dynamics, 19–20, 305