Adaptive Differential Evolution: A Robust Approach to Multimodal Problem Optimization 9783642015267, 9783642107009, 9783642128332, 9783642128684, 9783642128691, 3642015263

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Anyong Qing and Ching Kwang Lee Differential Evolution in Electromagnetics

Adaptation, Learning, and Optimization, Volume 4 Series Editor-in-Chief Meng-Hiot Lim Nanyang Technological University, Singapore E-mail: [email protected] Yew-Soon Ong Nanyang Technological University, Singapore E-mail: [email protected] Further volumes of this series can be found on our homepage: springer.com Vol. 1. Jingqiao Zhang and Arthur C. Sanderson Adaptive Differential Evolution, 2009 ISBN 978-3-642-01526-7 Vol. 2. Yoel Tenne and Chi-Keong Goh (Eds.) Computational Intelligence in Expensive Optimization Problems, 2010 ISBN 978-3-642-10700-9 Vol. 3. Ying-ping Chen (Ed.) Exploitation of Linkage Learning in Evolutionary Algorithms, 2010 ISBN 978-3-642-12833-2 Vol. 4. Anyong Qing and Ching Kwang Lee Differential Evolution in Electromagnetics, 2010 ISBN 978-3-642-12868-4

Anyong Qing and Ching Kwang Lee

Differential Evolution in Electromagnetics

123

Dr. Anyong Qing Temasek Laboratories National University of Singapore 5 Sports Dr 2 Singapore 117508 E-mail: [email protected]

Dr. Ching Kwang Lee School of Electrical and Electronic Engineering Nanyang Technological University Division of Communication Engineering S1-B1a-10, Nanyang Avenue Singapore 639798 E-mail: [email protected]

ISBN 978-3-642-12868-4

e-ISBN 978-3-642-12869-1

DOI 10.1007/978-3-642-12869-1 Adaptation, Learning, and Optimization

ISSN 1867-4534

Library of Congress Control Number: 2010926029 c 2010 Springer-Verlag Berlin Heidelberg 

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 987654321 springer.com

Preface

1 Motivations After many years of development and applications, differential evolution has proven itself a very simple while very powerful stochastic global optimizer. Since its inception, it has been applied to solve problems in many scientific and engineering fields. Nowadays, our daily life relies heavily on electromagnetics. Differential evolution has played an essential role in many synthesis and design problems in electromagnetics. This book focuses on applications of differential evolution in electromagnetics to showcase the achievement of differential evolution and further boost its acceptance in electromagnetics community.

2 Layout This book is composed of two parts. Part one includes the first three chapters while the remaining five chapters belong to part two of this book. 2.1

Part One

This part focuses on a literature survey on differential evolution. As far as we know, it is by far the most extensive and exhaustive one. 2.1.1 Chapter 1 Chapter 1 gives details of the literature survey which covers publication collection, refining and analysis. It opens up with the purposes this literature survey aims to serve. Next, Platforms over which the literature survey is actually conducted are then discussed. Initial statistical results over these platforms are presented. After that, the refining process to remove irrelevant publications is discussed. Yearly outputs of formal publications with and without refining are presented. Result analysis, or publication classification, is then discussed. Topics according to which collected publications are clustered are suggested. In particular, theoretical studies on differential evolution are summarized. Finally, some future actions are discussed. We have noticed several misconceptions and misconducts on differential evolution through this literature survey. They are clearly pointed out at the end of this chapter.

VI

Preface

2.1.2 Chapter 2 Basics of differential evolution are presented in Chapter 2. It also serves the second part of this book so that repetition of description of differential evolution is avoided. A short history of differential evolution is first discussed. It covers its inception, early years until 1998 and years from 1998 onwards. Major events in early years and key milestones in years from 1998 onwards are highlighted. The basic framework of differential evolution is then explained by revisiting the originators’ inventory publication, followed by a description of the more generic classic differential evolution. Some prominent variants of the two fundamental evolutionary operations in differential evolution are presented. Next, dynamic differential evolution which was misunderstood and seriously underestimated before is briefly mentioned. Finally, essential features of differential evolution including both advantages and disadvantages are highlighted. It has to be pointed out that a state of the art review of differential evolution is not presented in this book due to tight time limit. Such a review will be part of a forming up encyclopedia of differential evolution. 2.1.3 Chapter 3 A retrospection of applications of differential evolution in electromagnetics in and before 2008 is presented in this Chapter. The coverage of the retrospection is clearly specified right at the beginning of this Chapter. The pioneering works of applications of differential evolution in electromagnetics are highlighted. Statistical results by both publication year and subject are presented. Detailed discussion of applications of differential evolution in specific subject is then given. Involved subjects include electromagnetic inverse problems, antenna arrays, microwave & RF engineering, antennas, electromagnetic structures, electromagnetic composite materials, frequency planning, radio network design, MIMO, radar, computational electromagnetics and electromagnetic compatibility. An outlook of applications of differential evolution in electromagnetics is also presented at the end of this Chapter. 2.2

Part Two

This part presents five new applications of differential evolution in differential evolution by different research groups. 2.2.1 Chapter 4 Reconstruction of two-dimensional dielectric cylinders by using differential evolution is presented in this Chapter. The efficiency of differential evolution has been numerically shown through various examples. In addition, the impact of initial guess on differential evolution is presented. The multiple signal classification is used to determine the number of cylinders, their approximate centers and approximate geometric dimensions while a least squares based method is used to generate an estimate of the permittivity of the cylinders. It has been shown that a proper choice of the initial guess can speed up the convergence of the optimization significantly.

Preface

VII

2.2.2 Chapter 5 Inspection of penetrable objects by using differential evolution together with a recently proposed iterative multiscaling approach is discussed in this Chapter. The solving procedure starts from a fixed test area and successively focuses on one or more "regions of interest" in order to determine the approximate shapes of the unknown objects. At each step of the minimization process, differential evolution is used to retrieve this support by minimizing a proper functional, which relates the measured scattered field data to the data numerically produced, at any iteration, by the current solution. Several new results are included concerning the reconstruction of inhomogeneous targets under various imaging conditions. The combined strategy has been proved to be quite effective in reconstructing complex dielectric cylinders such as hollow and E-shape cylinders in noisy environment. 2.2.3 Chapter 6 In this Chapter a flexible method for prediction of far-field radiated emissions is presented. It is a promising computational alternative to the expensive large semianechoic chambers necessary to perform electromagnetic compatibility far-field radiated emission measurements. In this method, the equipment under test is replaced by an equivalent set of infinitesimal dipoles (both electric and magnetic) distributed inside the volume occupied by the equipment under test which is determined from near-field measurements at a short distance of the equipment under test. A memetic metaheuristic technique combing genetic algorithms, differential evolution and downhill simplex method is used to determine the type, position, orientation and excitation current of each dipole of the equivalent set of dipoles. The information obtained from the equivalent dipole set is used to determine the radiation at the far-field, as well as to identify the radiating parts of the equipment. 2.2.4 Chapter 7 Differential evolution with Pareto tournaments (DEPT) was applied to address the multi-objective optimization of frequency assignment problem in two real-world GSM networks in this Chapter. Two performance indicators, hypervolume and coverage relation, are implemented to analyze results. Results are compared with those by other multi-objective metaheuristics. Final results show that fine-tuned DEPT outperforms both MO-VNS and MO-SVNS while performs worse than both GMO-SVNS and GMO-VNS, among which GMO-SVNS performs best. 2.2.5 Chapter 8 In this Chapter, differential evolution is combined with particle swarm optimization (PSO) and another evolutionary algorithm (EA) to create a novel hybrid algorithm, the PSO-EA-DEPSO. The alteration between PSO, PSO-EA, and DEPSO provides additional diversity to counteract premature convergence. This hybrid algorithm is then shown to outperform PSO, PSO-EA, and DEPSO when applied to wireless MIMO channel prediction.

VIII

Preface

3 Readership As its name indicates, this book is specially prepared for electromagnetic researchers facing optimization problems. It will be particularly attractive to researchers who have been frustrated by other optimization algorithms. This book is a premium resource for differential evolution community. People in this community will have a better understanding on differential evolution and its huge application potential. This book is also an ideal resource for evolutionary computation community. People in this community may find it helpful in presenting a more appropriate approach to conduct concerned literature survey and providing real engineering application examples.

Acknowledgement

First of all, I would take this opportunity to thank Prof. Hock Lim and Mr. Joseph Sing Kwong Ting, directors of Temasek Laboratories, National University of Singapore, for their support and encouragement of my study on differential evolution. The financial funding from Defence Science & Technology Agency, Singapore is greatly appreciated too. I would also like to thank Prof. Meng-Hiot Lim, series editor on evolutionary learning and optimization, and Dr. Thomas Ditzinger, senior editor within Springer Verlag responsible for this series, for their strong recommendation to publish this book. Thanks also go to Mr. Heather King for his carefulness and patience.

Contents

Contents 1

A Literature Survey on Differential Evolution…………………………...1 1.1 Motivations .............................................................................................1 1.1.1 Eliminating Inconsistencies .........................................................1 1.1.2 Crediting Original Contributions .................................................1 1.1.3 Knowing the State of the Art .......................................................1 1.1.4 Gaining Insight ............................................................................2 1.2 Platforms.................................................................................................2 1.2.1 Starting Point ...............................................................................2 1.2.2 Databases .....................................................................................3 1.2.3 Informal Online Resources and Tools .........................................4 1.3 Result Refining .......................................................................................5 1.3.1 Books ...........................................................................................5 1.3.2 Book Chapters .............................................................................6 1.3.3 Other Formal Publications ...........................................................6 1.3.4 Informal Notes .............................................................................7 1.4 Result Analysis .......................................................................................7 1.4.1 Theory of Differential Evolution .................................................7 1.4.2 Fundamentals of Differential Evolution ......................................8 1.4.3 Intrinsic Control Parameters ........................................................9 1.4.4 Evaluation of Differential Evolution............................................9 1.4.5 Applications of Differential Evolution ........................................9 1.4.6 Hybridization ...............................................................................9 1.5 Future Actions ......................................................................................10 1.5.1 Open Access ..............................................................................10 1.5.2 Future Update ............................................................................10 1.6 Misconceptions and Misconducts on Differential Evolution................10 References .............................................................................................................10

2

Basics of Differential Evolution…………………………………………..19 2.1 A Short History.....................................................................................19 2.1.1 Inception ....................................................................................19 2.1.2 Early Years ................................................................................20 2.1.2.1 Assessment..................................................................20 2.1.2.2 Reputation Building ....................................................20

XII

Contents

2.2

2.3

2.4 2.5 2.6

2.1.2.3 Applications ................................................................20 2.1.2.4 Promotion....................................................................21 2.1.2.5 Practical Advice ..........................................................21 2.1.2.6 Standardization ...........................................................22 2.1.2.7 More Adventures ........................................................22 2.1.3 Key Milestones in and after 1998 ..............................................22 The Foundational Differential Evolution Strategies .............................23 2.2.1 Notations....................................................................................23 2.2.2 Strategy Framework...................................................................24 2.2.2.1 Pseudo-code ................................................................24 2.2.2.2 Initialization ................................................................25 2.2.2.3 Differential Mutation ..................................................25 2.2.2.4 Crossover ....................................................................26 2.2.2.5 Selection......................................................................27 2.2.2.6 Termination Conditions ..............................................27 2.2.3 Intrinsic Control Parameters ......................................................27 Classic Differential Evolution ..............................................................28 2.3.1 Initialization ...............................................................................28 2.3.2 Differential Mutation .................................................................28 2.3.2.1 Current ........................................................................29 2.3.2.2 Best .............................................................................29 2.3.2.3 Better...........................................................................29 2.3.2.4 Random.......................................................................29 2.3.2.5 Mean ...........................................................................29 2.3.2.6 Best of Random ..........................................................29 2.3.2.7 Arithmetic Best ...........................................................29 2.3.2.8 Arithmetic Better ........................................................29 2.3.2.9 Arithmetic Random.....................................................29 2.3.2.10 Trigonometric .............................................................30 2.3.2.11 Directed.......................................................................30 2.3.3 Crossover ...................................................................................30 2.3.3.1 Binary Crossover ........................................................31 2.3.3.2 One-Point Crossover ...................................................32 2.3.3.3 Multi-point Crossover .................................................32 2.3.3.4 Arithmetic Crossover ..................................................32 2.3.3.5 Arithmetic One-Point Crossover.................................33 2.3.3.6 Arithmetic Multi-point Crossover...............................33 2.3.3.7 Arithmetic Binomial Crossover ..................................34 2.3.3.8 Arithmetic Exponential Crossover..............................35 Dynamic Differential Evolution ...........................................................36 State of the Art of Differential Evolution .............................................36 Essential Features of Differential Evolution.........................................37 2.6.1 Advantages ................................................................................37 2.6.1.1 Reliability....................................................................37 2.6.1.2 Efficiency....................................................................37

Contents

XIII

2.6.1.3 Simplicity....................................................................37 2.6.1.4 Robustness ..................................................................38 2.6.2 Disadvantages ............................................................................38 2.6.2.1 Efficiency....................................................................38 2.6.2.2 Incapability for Epistatic and Noisy Problems............38 References .............................................................................................................38 3

A Retrospective of Differential Evolution in Electromagnetics……….43 3.1 Introduction ..........................................................................................43 3.1.1 Coverage ....................................................................................43 3.1.2 Pioneering Works ......................................................................44 3.1.3 An Overview of Applications of Differential Evolution in Electromagnetics........................................................................44 3.1.3.1 Yearly Output .............................................................44 3.1.3.2 Output by Subject .......................................................44 3.2 Electromagnetic Inverse Problems .......................................................45 3.2.1 A Bird’s Eye View.....................................................................45 3.2.2 Further Classification.................................................................45 3.2.2.1 One-Dimensional Electromagnetic Inverse Problems .....................................................................46 3.2.2.2 Two-Dimensional Electromagnetic Inverse Problems .....................................................................46 3.2.2.3 Three-Dimensional Electromagnetic Inverse Problems .....................................................................47 3.3 Antenna Arrays.....................................................................................48 3.3.1 Conventional Antenna Arrays....................................................48 3.3.1.1 Ideal Antenna Arrays ..................................................48 3.3.1.2 Practical Antenna Arrays ............................................48 3.3.1.3 Phased Arrays .............................................................49 3.3.2 Time-Modulated Antenna Arrays ..............................................49 3.3.2.1 Ideal Antenna Arrays with Time Modulation .............49 3.3.2.2 Practical Antenna Arrays with Time Modulation .......49 3.3.2.3 Phased Antenna Arrays with Time Modulation..........50 3.3.3 Moving Phase Center Antenna Arrays.......................................50 3.4 Microwave and RF Engineering ...........................................................50 3.4.1 Design of Microwave and RF Devices ......................................50 3.4.1.1 Designing Microwave and RF Devices Using Differential Evolution .................................................51 3.4.1.2 Extracting Empirical Synthesis Formulas Using Differential Evolution .................................................51 3.4.2 Characterization of Microwave and RF Devices .......................51 3.4.2.1 Calibration of Measuring System for Characterizing Microwave and RF Devices ........................................51 3.4.2.2 Modeling of Microwave and RF Devices ...................52 3.5 Antennas ...............................................................................................52 3.5.1 Design of Antennas....................................................................52

XIV

Contents

3.5.1.1 Designing Antennas Using Differential Evolution .....52 3.5.1.2 Extracting Empirical Formulas for Synthesizing Antennas .....................................................................52 3.5.2 Measurement of Antennas .........................................................53 3.6 Electromagnetic Structures ...................................................................53 3.6.1 Plain Electromagnetic Structures ...............................................54 3.6.2 Frequency Selective Surfaces ....................................................55 3.7 Electromagnetic Composite Materials ..................................................56 3.7.1 Modeling of Electromagnetic Composite Materials ..................56 3.7.2 Retrieval of Effective Permittivity Tensor.................................56 3.8 Frequency Planning ..............................................................................57 3.9 Radio Network Design..........................................................................58 3.10 MIMO...................................................................................................58 3.11 Radar.....................................................................................................59 3.12 Computational Electromagnetics ..........................................................59 3.13 Electromagnetic Compatibility .............................................................60 3.14 Miscellaneous Applications ..................................................................60 3.15 An Outlook to Future Applications of Differentia Evolution in Electromagnetics...................................................................................60 References .............................................................................................................61

4

Application of Differential Evolution to a Two-Dimensional Inverse Scattering Problem …………………………......………………………...73 4.1 Introduction ..........................................................................................73 4.2 General Description of the Problem .....................................................74 4.2.1 Experimental Setup....................................................................74 4.2.2 The Optimization Problem.........................................................76 4.3 Mathematical Nature of the Optimization Problem and Differential Evolution ..............................................................................................76 4.4 Initial Guess ..........................................................................................77 4.4.1 Foldy-Lax Model of Scattering..................................................78 4.4.2 Multiple Signal Classification for Estimating the Scatterer Support.......................................................................................79 4.4.3 Least Square Based Method for Generating Initial Guess for the Relative Permittivity ..................................................................80 4.5 Numerical Results.................................................................................81 4.5.1 Measurement Setup....................................................................81 4.5.2 Control Parameters ....................................................................81 4.5.3 Numerical Example 1: A Single Cylinder .................................82 4.5.4 Numerical Example 2: Two Identical Cylinders........................87 4.5.5 Numerical Example 3: Two Different Cylinders .......................90 4.5.6 Numerical Example 4: Two Closely Located Identical Cylinders....................................................................................94 4.5.7 Numerical Example 5: Kite Cross-Section Cylinder .................97

Contents

XV

4.6 Conclusions ........................................................................................101 References ...........................................................................................................102 5

The Use of Differential Evolution for the Solution of Electromagnetic Inverse Scattering Problems…………………………………………….107 5.1 Introduction ........................................................................................107 5.2 Problem Formulation ..........................................................................108 5.2.1 The Inverse Scattering Formulation.........................................108 5.2.2 Discrete Setting........................................................................109 5.2.3 The Inverse Scattering Problem as an Optimization Problem....................................................................................110 5.3 The Iterative Multiscaling Approach ..................................................110 5.4 Numerical Results...............................................................................112 5.4.1 Off-Centered Dielectric Cylinder ............................................112 5.4.2 Off-Centered Dielectric Hollow Cylinder................................117 5.4.3 Centered Stratified Dielectric Square Cylinder........................121 5.4.4 Centered E-Shape Dielectric Cylinder .....................................126 5.5 Conclusions ........................................................................................129 References ...........................................................................................................129

6

Modeling of Electrically Large Equipment with Distributed Dipoles Using Metaheuristic Methods …………………………………………..133 6.1 Introduction ........................................................................................133 6.1.1 Near-Field to Far-Field Transformation ..................................133 6.1.2 Radiating Equipment Modeling with Prefixed Position Dipoles.....................................................................................134 6.1.3 Present Work ...........................................................................135 6.2 Electromagnetic Modeling of a Radiating Equipment with Distributed Infinitesimal Dipoles ..........................................................................135 6.2.1 Integral Equations for the Radiation of Electronic Equipment................................................................................136 6.2.2 Point-Matching Method with Dirac Delta Basis Functions .....137 6.2.3 Ground Plane in Semi-anechoic Chambers..............................137 6.3 Proposed Method for Near-Field to Far-Field Transformation...........138 6.3.1 Description of the Method .......................................................138 6.3.2 Optimization Problem..............................................................140 6.3.3 Source Identification................................................................140 6.4 Electromagnetic Optimization by Genetic Algorithms.......................140 6.4.1 EMOGA v1.0: Genetic Algorithm...........................................141 6.4.2 EMOGA 2.0: Metaheuristic Method .......................................142 6.4.2.1 Current Scaling .........................................................142 6.4.2.2 Correlation between Dipoles.....................................143 6.4.2.3 Memetization ............................................................143 6.5 Numerical Results...............................................................................144 6.5.1 Measurement Systems .............................................................144 6.5.1.1 General Measurement System ..................................144

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Contents

6.5.1.2 Near-Field Measurement System..............................145 6.5.1.3 Far-Field Measurement System ................................147 6.5.2 Near-Field Results ...................................................................147 6.5.3 Far-Field Prediction ............................................................... `149 6.6 Conclusions ........................................................................................150 References ...........................................................................................................151 7

Application of Differential Evolution to a Multi-Objective Real-World Frequency Assignment Problem………………………………………...155 7.1 Introduction ........................................................................................155 7.2 Multi-objective FAP in a GSM Network............................................156 7.2.1 GSM Components and Frequency Planning ............................156 7.2.2 Interference Cost......................................................................157 7.2.3 Separation Cost ........................................................................158 7.3 Multi-objective Differential Evolution with Pareto Tournaments ......159 7.3.1 Algorithm Structure .................................................................159 7.3.2 Pareto Tournament...................................................................159 7.3.3 Problem Domain Knowledge...................................................160 7.4 Multi-objective Variable Neighborhood Search .................................160 7.4.1 Variable Neighborhood Search................................................160 7.4.2 Multi-objective Variable Neighborhood Search ......................161 7.4.3 Greedy Mutation ......................................................................162 7.4.4 Multi-objective Skewed Variable Neighborhood Search.........162 7.5 Experiments and Results.....................................................................163 7.5.1 Experimental Setup..................................................................163 7.5.1.1 Used GSM Instances.................................................163 7.5.1.2 Encoding ...................................................................165 7.5.1.3 Computational Facilities ...........................................165 7.5.1.4 Termination Conditions and Process Monitoring .....165 7.5.1.5 Confidence Building .................................................165 7.5.2 Methodology and Metrics ........................................................166 7.5.2.1 Hypervolume ............................................................166 7.5.2.2 Coverage Relation.....................................................166 7.5.3 Tuning of the DEPT Parameters ..............................................166 7.5.3.1 Population Size .........................................................167 7.5.3.2 Crossover Probability................................................168 7.5.3.3 Mutation Intensity.....................................................169 7.5.3.4 DEPT Scheme...........................................................170 7.5.3.5 Findings ....................................................................172 7.5.4 Empirical Results.....................................................................173 7.6 Conclusions ........................................................................................174 References ...........................................................................................................175 8

RNN Based MIMO Channel Prediction……………..............................177 8.1 Introduction ........................................................................................177 8.2 Received Signal Model.......................................................................178

Contents

XVII

8.2.1 Received Signal Model ............................................................178 8.2.2 Optimization Problem..............................................................179 8.3 Hybrid PSO-ES-DEPSO Training Algorithm.....................................179 8.4 MIMO Channel/Beam-Forming Models ............................................180 8.4.1 Channel Model.........................................................................180 8.4.2 Channel Estimation Model ......................................................182 8.4.3 MIMO Beam-Forming.............................................................182 8.5 Recurrent Neural Network for Channel Prediction.............................184 8.6 Training Procedure .............................................................................185 8.7 Numerical Results...............................................................................187 8.7.1 Algorithm Comparison ............................................................187 8.7.2 Robustness of PSO-ES-DEPSO Algorithm .............................188 8.7.3 Linear and Nonlinear Predictors with PSO-EA-DEPSO Algorithm.................................................................................191 8.7.4 Non-convexity of the Solution Space ......................................192 8.8 Performance Measures of RNN Predictors.........................................193 8.9 Conclusions ........................................................................................203 References ...........................................................................................................204 Index……….…………………………………………………………………...207

Chapter 1

A Literature Survey on Differential Evolution Anyong Qing

1

1.1 Motivations 1.1.1 Eliminating Inconsistencies It has been observed since 2004 that there are many inconsistent or even false claims prevailing in the community of differential evolution [1]. Two measures have been taken to clarify them. The first is a system level parametric study on differential evolution [1]-[4]. The second is the large scale literature survey mentioned here. It is one of the foundation stones of this book.

1.1.2 Crediting Original Contributions The academic society nowadays has become more and more utilitarian and impetuous. Many researchers dream a shortcut to their academic success. They tend to accept established view points especially those from topical review articles by leading researchers. Original publications are neglected that insufficient credits are given to originality. In some cases, they may not be aware that the original contributions are cited incorrectly [1]. Academic misconducts such as multiple submissions, exaggerated claims, or even plagiarism are not rare. It is one of the objectives of this survey to promote good academic conducts by locating and appropriately crediting original contributions.

1.1.3 Knowing the State of the Art It has been more than ten years since the inception of differential evolution. However, as far as we know, nobody else has done any comprehensive literature Anyong Qing Temasek Laboratories, National University of Singapore 5A, Engineering Dr 1 #06-09, Singapore 117411 e-mail: [email protected]

1

A. Qing and C.K. Lee: Differential Evolution in Electromagnetics, ALO 4, pp. 1–17. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

2

A. Qing

survey on differential evolution. The state of the art of differential evolution is therefore not precisely known to interested researchers. This literature survey aims to fill this gap. It also serves to reveal the popularity of differential evolution.

1.1.4 Gaining Insight The literature survey involves not only literature collection but also literature analysis among which the latter is more important. Through the analysis, the following questions will be answered (a) (b) (c)

What is differential evolution? When is differential evolution used and why is it useful? When will differential evolution fail and why does it fail?

Answers to the above questions are crucial for potential applications of differential evolution. Insights gained may lead to future improvements on differential evolution.

1.2 Platforms In general, there are two platforms to look for publications on differential evolution. Although conventional publications printed on paper still play an important role, digital resources electronically available have been increasingly more preferred by both researchers and publishers. We have seen a quick transition from paper platform to digital platform within the last decade. It is noticed that differential evolution was proposed when paper platform was dominating [1], [5]-[15]. However, the dominance of paper platform does not last long. Both researchers and publishers have quickly realized the advantages of digital platform and have not hesitated to turn their attention to it. In this regard, digital platform is chosen as the main platform to carry out the literature survey. However, it is not the sole platform. Paper platform is also implemented whenever possible to supplement the digital platform so that missing of publications is minimized.

1.2.1 Starting Point At the initial stage, the literature survey is selective. Attention was focused on a bibliography [16] compiled by Prof. J. A. Lampinen which was posted online for open access. The bibliography itself was downloaded. Each and every publication included was also downloaded or copied whenever possible. The bibliography was expanded to include missing relevant publications appearing as references in available publications in the bibliography. Unfortunately, the bibliography has not been updated since it was last updated on Oct. 14, 2002. Consequently, it is very incomplete and can not stand in line with the state of the art of differential evolution.

1 A Literature Survey on Differential Evolution

3

1.2.2 Databases Many organizations have established their own databases which have been subscribed by most major libraries. Publications covered are usually formal, in another word, peer-reviewed and printed on paper. Nowadays, these databases go electronic for more exposure. They are also more timely updated. A publication from the following databases is counted if the keyword, differential evolution, appears in any field (title, abstract, keywords, text, and references) in the publication. Number of hits from different databases is shown in Table 1.1. (a) (b) (c) (d) (e) (f) (g) (h) (i)

Chinese Electronic Periodical Services Engineering Village 2 (EI) IEEE Explore Institute of Scientific and Technical Information of China ISI Web of Science (SCI) National Knowledge Infrastructure Scopus & ScienceDirect SpringerLink Wiley InterScience Table 1.1 Number of Hits from Different Databases year

IEEE

SCI

EI

InterScience

SpringerLink

Scopus

1995

0

7

9

0

1

4

1996

4

3

10

2

0

56

1997

4

9

19

2

7

56

1998

4

5

12

0

7

51

1999

8

18

45

7

7

71

2000

10

25

43

8

10

93

2001

8

23

50

4

11

112

2002

18

37

56

3

14

130

2003

23

60

96

8

15

206

2004

32

76

157

10

30

302

2005

55

132

247

11

43

381

2006

88

156

343

12

131

542

2007

116

258

528

26

143

696

2008

209

384

768

35

195

1132

4

A. Qing

Please note that (a) The last search was conducted in October, 2009. To avoid any potential misleading to readers, partial search result for year 2009 is not presented here. (b) The search results may contain irrelevant publications on differential evolution equation, social differential evolution, cultural differential evolution, economical differential evolution, geological differential evolution, geographical differential evolution, genetic differential evolution, and so on. (c) Each and every database has its unique coverage. No database is exhaustive. (d) Usually, a database contains publications from the publisher owning the database. However, Scopus provided by Elsevier covers some non-Elsevier publications. Therefore, the number of hits on Scopus is the largest except for year 1995. In this sense, Scopus is more comprehensive. (e) The search is focused on publications in Chinese and English. Publications presented in other languages are not considered unless they are indexed in the above databases.

1.2.3 Informal Online Resources and Tools Besides the above electronic resources, many informal electronic resources scatter over the internet. Some of the covered publications are notes that have not been published in any formal publishing platforms such as books, journals, conference proceedings, technical reports, or theses. They are usually posted to the internet by either individual researchers or non-academic and/or non-profitable organizations for various purposes. Access is in general free. These resources can be reached with the help of free search engines such as Google, Yahoo, Microsoft Bing, or Ask (http://www.ask.com/). Alternatively, researchers may visit the websites where these resources are actually stored. Three of the most prominent websites are Google Scholar (http://scholar.google.com.sg/) provided by Google, Computer Science Bibliographies (http://liinwww.ira.uka.de/bibliography/index.html) maintained by AlfChristian Achilles and Paul Ortyl, and citeSeerX (http://citeseerx.ist.psu.edu/) provided by College of Information Sciences and Technology, Pennsylvania State University. The number of hits from these three websites is shown in Table 1.2. The search result is accurate as of October 22, 2009 and may similarly contain irrelevant publications. Likewise, partial search result for year 2009 is not presented here.

1 A Literature Survey on Differential Evolution

5

Table 1.2 Number of Hits from Google Scholar, CSB, and citeSeerX

Google Scholar

CSB

citeSeerX

1995

6

9

6

1996

5

10

6

1997

6

6

6

1998

7

6

8

1999

38

18

9

2000

10

20

12

2001

12

21

17

2002

17

27

20

2003

24

32

22

2004

43

80

24

2005

67

80

24

2006

92

103

19

2007

138

167

9

2008

206

220

7

Due to the vast number of publications available from the internet, the search for relevant publications can only be done in a very restrictive way. Collection of publications here is accordingly far less than exhaustive.

1.3 Result Refining As mentioned before, publications found by the search may be irrelevant. Therefore, refining the survey result to eliminate irrelevant publications is compelling. This is done through tediously reading the reachable part of each and every found publication. Qualified publications are classified into four major categories: books, book chapters, other formal publications, and informal publications.

1.3.1 Books Book is a very important form of publications. 5 monographs on differential evolution [1], [17]-[20] have been published by now, among which the first book by the originators is introductory and well circulated.

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1.3.2 Book Chapters Some of the publications are presented as chapters in edited books. By now, at least one chapter is dedicated to differential evolution in at least 86 books [21]-[106] before (excluding) 2009.

1.3.3 Other Formal Publications Publications covered here include journal papers, presentations in conferences, symposiums, and workshops, degree theses, and technical reports. The number of qualified publications under this category is shown graphically in Fig. 1.1 where the ordinate is the year of publication while the number of qualified publications under this category is given beside the corresponding bar. Books and book chapters are excluded. Steady and accelerating growth has been observed since 1995, in which differential evolution was proposed. 1029

2008 744

2007 2006

536 363

2005 236

2004 2003

145 94

2002

79

2001 2000

57

1999

54

1998

21 16

1997 1996

7

1995

4

Fig. 1.1 Formal Publications on Differential Evolution

It has been noted that some publications do not actually involve a case study on differential evolution. Such publications do not contribute anything to differential evolution and are therefore unimportant to the differential evolution community. No further analysis on these publications is necessary. Remaining publications is shown graphically in Fig. 1.2. Similarly, books and book chapters are excluded. Please note that some publications may be wrongly treated as those without case studies on differential evolution because of availability.

1 A Literature Survey on Differential Evolution

7

638

2008 507

2007 365

2006 257

2005 154

2004 99

2003 68

2002

62

2001 41

2000

48

1999 1998

17

1997

15

1996

6

1995

4

Fig. 1.2 Formal Publications with Case Study on Differential Evolution

1.3.4 Informal Notes All qualified publications outside the above three categories are assembled under this category. There are 6 publications in this category as of October 22, 2009. Most of them are preprints or PowerPoint presentation notes posted to the internet by individual researchers. Bibliographic record is incomplete.

1.4 Result Analysis To make differential evolution benefit more existing researchers from active application fields and attract hesitating researchers from promising application fields, analysis on collected publications has to be conducted. The main goal of the result analysis is to look for practical usage advice for future applications and gain insight to further improve differential evolution. The analysis assembles publications on a specific topic so that researchers interested in the topic will not waste their time on irrelevant publications. At present, the analysis is focused on the following topics.

1.4.1 Theory of Differential Evolution Building a precise mathematical model for differential evolution has been posed as a challenge to the community as soon as differential evolution was proposed

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[1]. Such a solid mathematical foundation, if any, may hint desired revolutionary upgrading of differential evolution for higher reliability, better efficiency, and more robustness. However, it is yet to establish. Theoretical treatments on differential evolution in the history of differential evolution are very rare. The condition in rigorous mathematics under which differential evolution is sure to converge, the most fundamental question facing the differential evolution community or even the whole evolutionary computation community, is still pending for answer. Mathematical models for involved evolutionary operations have not been built. Interaction between evolutionary operations, intrinsic and non-intrinsic control parameters, and problems features has not been disclosed either. There is still a long way to go before differential evolution is fully appreciated. Nevertheless, some valuable pioneering efforts have been made to have a deeper understanding on differential evolution and may lead to the eventual theory of differential evolution. Studies by the originators are presented in their introductory monograph [17] and the latest book chapter by Price [19]. Other relevant studies are summarized in Table 1.3.

Table 1.3 Studies on Theory of Differential Evolution

focus

reference

evolution dynamics

[107]-[108]

stagnation

[109]-[111]

differential mutation

[112]-[115]

crossover

[116]-[117]

selection

[116]

parameter adaptation

[118]-[119]

Termination conditions

[120]-[125]

1.4.2 Fundamentals of Differential Evolution It is very critical for an applicant of differential evolution to have sufficient knowledge on differential evolution so that he or she can choose the most suitable differential evolution strategy and its corresponding intrinsic control parameter values. Otherwise, he or she may be confused by the huge number of differential evolution strategies and the infinite possibilities of setting intrinsic control parameter values. In the worst scenario, he or she may even be misled by past

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inappropriate usage of differential evolution and get an unnecessarily negative impression on differential evolution. An analysis on collected publications regarding fundamentals of differential evolution is in progress. Subjects in mind include evolution mechanism, encoding and decoding, initialization, differential mutation, crossover, selection, termination conditions, constraint handling, co-evolution, and so on.

1.4.3 Intrinsic Control Parameters Pleasant usage of differential evolution comes with not only the most suitable strategy but also appropriate setting of intrinsic control parameters. It has been well known that intrinsic control parameters of differential evolution play an essential role. Publications focusing on studying intrinsic control parameters will be assembled separately.

1.4.4 Evaluation of Differential Evolution It is a common practice to find out the advantages and disadvantages of an optimization algorithm through evaluation and comparison. Differential evolution has been evaluated by many researchers over various test bed and has earned its reputation in many comparative evaluations. Through specific evaluations, we may figure out a clearer picture on concerned component of differential evolution.

1.4.5 Applications of Differential Evolution Differential evolution will eventually be applied to solve practical application problems. Past applications provides valuable experience on usage of differential evolution. A preliminary attempt to classify qualified publications has been made to identify the fields in which differential evolution has been applied [1]. Such a preliminary classification did not cover publications in and after year 2008 and may be imprecise due to limited personal knowledge.

1.4.6 Hybridization It has been a well known fact that “for any algorithm, any elevated performance over one class of problems is offset by performance over another class” [126]. In another word, an optimization algorithm may outperform its counterparts over a specific class of problems. There is no exception for differential evolution. In accordance, many researchers have tried to hybridize different differential evolution strategies or differential evolution with other optimization algorithms. Analyzing different hybridization approaches is now in progress.

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1.5 Future Actions 1.5.1 Open Access We are currently managing a personal library covering different fields, among which differential evolution is one of the essential components. The personal library will be posted to internet for open access at appropriate time. It has to be seriously pointed out that although we have taken every possible measure to minimize missing publications, exhaustiveness of collection can not be claimed now and will not be claimed at any time in the future. Feedback from readers about any missing publications, be them formal or informal, is always welcome and appreciated. Missing publications notified by informants will be immediately integrated into the library. Moreover, this author will never claim absolute accuracy for the collection and analysis results. Besides limited personal knowledge, availability of collected publications may be the prime culprit. By chance, some title-only publications are collected during the search process. This author sincerely appeals to publishers, authors, and readers having such publications to share them as much as possible among the differential evolution community.

1.5.2 Future Update Finalizing for year 2009 will be carried out in early 2010. Update for years 2010 onwards is expected to take place on a yearly basis in order to make the survey result more accurate.

1.6 Misconceptions and Misconducts on Differential Evolution Differential evolution is one of the essential members of evolutionary algorithms. It shares some evolutionary operations and/or essential features with other evolutionary algorithms. However, it does not mean that it can be used interchangeably with other evolutionary algorithms. It is fundamentally different with other evolutionary algorithms in terms of evolution mechanism and evolutionary operations even if some evolutionary operations in differential evolution are identically named for historical reasons. It has been noticed that differential evolution has been mistermed as differential genetic algorithm [127] and has been regarded as variations of genetic algorithms [128]-[129], evolution strategies [130]. Such terms and/or classification are misleading.

References [1] Qing, A.: Differential Evolution: Fundamentals and Applications in Electrical Engineering. John Wiley, New York (2009) [2] Qing, A.: Dynamic differential evolution strategy and applications in electromagnetic inverse scattering problems. IEEE Trans. Geosci. Remote Sens. 44(1), 116– 125 (2006)

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[3] Qing, A.: A parametric study on differential evolution based on benchmark electromagnetic inverse scattering problem. In: 2007 IEEE Congress Evolutionary Computation, Singapore, September 25-28, pp. 1904–1909 (2007) [4] Qing, A.: A study on base vector for differential evolution. In: 2008 IEEE World Congress Computational Intelligence/2008 IEEE Congress Evolutionary Computation, Hong Kong, June 1-6, pp. 550–556 (2008) [5] Storn, R.: Modeling and Optimization of PET-Redundancy Assignment for MPEGSequences, Technical Report TR-95-018, International Computer Science Institute (May 1995) [6] Storn, R.: Differential Evolution Design of an IIR-Filter with Requirements for Magnitude and Group Delay, Technical Report TR-95-026, International Computer Science Institute (June 1995) [7] Storn, R., Price, K.V.: Differential Evolution - A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces, Technical Report TR-95012, International Computer Science Insitute (Mar 1995) [8] Price, K.V.: Differential evolution: a fast and simple numerical optimizer. In: 1996 Biennial Conf. North American Fuzzy Information Processing Society, Berkeley, CA, June 19-22, pp. 524–527 (1996) [9] Storn, R.: Differential evolution design of an IIR-filter. In: 1996 IEEE Int. Conf. Evolutionary Computation, Nagoya, May 20-22, pp. 268–273 (1996) [10] Storn, R.: On the usage of differential evolution for function optimization. In: 1996 Biennial Conf. North American Fuzzy Information Processing Society, Berkeley, CA, June 19-22, pp. 519–523 (1996) [11] Storn, R.: System Design by Constraint Adaptation and Differential Evolution, Technical Report TR-96-039, International Computer Science Institute (November 1996) [12] Storn, R., Price, K.V.: Minimizing the real functions of the ICEC’96 contest by differential evolution. In: 1996 IEEE Int. Conf. Evolutionary Computation, Nagoya, May 20-22, pp. 842–844 (1996) [13] Price, K.V.: Differential evolution vs. the functions of the 2nd ICEO. In: 1997 IEEE Int. Conf. Evolutionary Computation, Indianapolis, IN, April 13-16, pp. 153–157 (1997) [14] Price, K., Storn, R.: Differential evolution: a simple evolution strategy for fast optimization. Dr. Dobb’s J. 22(4), 18–24, 78 (1997) [15] Storn, R., Price, K.V.: Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optimization 11(4), 341–359 (1997) [16] Lampinen, J.: A bibliography on differential evolution algorithm, Technical Report, Lappeenranta University of Technology, Department of Information Technology, Laboratory of Information Processing (2001) (last updated on October 14, 2002) available via internet, http://www2.lut.fi/~jlampine/debiblio.htm (accessed on October 12, 2009) [17] Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution: a Practical Approach to Global Optimization. Springer, Berlin (2005) [18] Feoktistov, V.: Differential Evolution: in Search of Solutions. Springer, Berlin (2006) [19] Chakraborty, U.K. (ed.): Advances in Differential Evolution. Springer, Berlin (2008)

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[20] Onwubolu, G.C., Davendra, D.: Differential Evolution: A Handbook for Global Permutation-Based Combinatorial Optimization-Studies in Computational Intelligence, vol. 175. Springer, Heidelberg (2009) [21] Corn, D., Dorigo, M., Glover, F. (eds.): New Ideas in Optimization. McGraw-Hill, London (1999) [22] Mastorakis, N.E. (ed.): Recent Advances in Circuits and Systems. World Scientific, Singapore (1998) [23] Topping, B.H.V. (ed.): Developments in computational mechanics with high performance computing. Civil-Comp Press, Edinburgh (1999) [24] Sincak, P., Vascak, J., Kvasnicka, V., Pospichal, J. (eds.): Intelligent Technologies Theory and Applications. IOS Press, Amsterdam (2002) [25] Huijsing, J.H., Steyaert, M., van Roermund, A. (eds.): Analog Circuit Design: Scalable Analog Circuit Design, High Speed D/A Converters, RF Power Amplifiers. Kluwer Academic Publishers, New York (2003) [26] Sarker, R., Mohammadian, M., Yao, X. (eds.): Evolutionary Optimization. Kluwer Academic Publishers, New York (2003) [27] Johnston, R.L. (ed.): Applications of Evolutionary Computation in ChemistryStructure & Bonding, vol. 110. Springer, Berlin (2004) [28] Onwubolu, G.C., Babu, B.V.: New Optimization Techniques in Engineering. Studies in Fuzziness and Soft Computing, vol. 141. Springer, Berlin (2004) [29] Zhong, J.J. (ed.): Biomanufacturing-Advances in Biochemical Engineering/Biotechnology, vol. 87. Springer, Berlin (2004) [30] Grigoras, D., Nicolau, A. (eds.): Concurrent information processing and computing. NATO science series, series III, Computer and systems sciences, vol. 195. IOS Press, Amsterdam (May 2005) [31] Hart, W.E., Krasnogor, N., Smith, J.E. (eds.): Recent Advances in Memetic Algorithms. Studies in Fuzziness and Soft Computing, vol. 166. Springer, Berlin (2005) [32] Hoffmann, F., Köppen, M., Klawonn, F., Roy, R. (eds.): Soft Computing: Methodologies and Applications-Advances in Soft Computing, vol. 32. Springer, Berlin (2005) [33] Palit, A.K., Popovic, D.: Computational Intelligence in Time Series Forecasting: Theory and Engineering Applications. Springer, Berlin (2005) [34] Pieruci, S. (ed.): Computer-Aided Chemical Engineering. Elsevier, Amsterdam (2005) [35] Tan, K.C., Khor, E.F., Lee, T.H.: Multiobjective Evolutionary Algorithms and Applications. Springer, Berlin (2005) [36] Abraham, A., de Baets, B., Köppen, M., Nickolay, B. (eds.): Applied Soft Computing Technologies: The Challenge of Complexity-Applied Soft Computing, vol. 34. Springer, Berlin (2006) [37] Abraham, A., Grosan, C., Ramos, V. (eds.): Stigmergic Optimization-Studies in Computational Intelligence, vol. 31. Springer, Berlin (2006) [38] Abraham, A., Grosan, C., Ramos, V. (eds.): Swarm Intelligence in Data Mining. Studies in Computational Intelligence, vol. 34. Springer, Berlin (2006) [39] Alba, E., Marti, R.: Metaheuristic Procedures for Training Neutral NetworksOperations Research/Computer Science Interfaces Series, vol. 36. Springer, Berlin (2006) [40] Brabazon, A., O’Neill, M.: Biologically Inspired Algorithms for Financial Modelling. Spriinger, Berlin (2006)

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[41] Burke, E.K., Kendall, G. (eds.): Search Methodologies: Introductory Tutorials in Optimization and Decision Support Techniques. Springer, Berlin (2006) [42] Caiti, A., Chapman, N.R., Hermand, J.P., Jesus, S.M. (eds.): Acoustic Sensing Techniques for the Shallow Water Environment: Inversion Methods and Experiments. Springer, Berlin (2006) [43] Castro-López, R., Fernández, F.V., Guerra-Vinuesa, O., Rodríguez-Vázquez, Á.: Reuse-Based Methodologies and Tools in the Design of Analog and Mixed-Signal Integrated Circuits. Springer, Berlin (2006) [44] Dzemyda, G., Šsltenis, V., Žilinskas, A. (eds.): Stochastic and Global Optimization. Springer, Berlin (2006) [45] Jin, Y. (ed.): Multi-Objective Machine Learning. Studies in Computational Intelligence, vol. 16. Springer, Berlin (2006) [46] Li, Z., Halang, W.A., Chen, G.: Integration of Fuzzy Logic and Chaos. Theory. Studies in Fuzziness and Soft Computing, vol. 187. Springer, Berlin (2006) [47] Liberti, L., Maculan, N. (eds.): Global Optimization: from Theory to Implementation-Nonconvex Optimization and Its Applications, vol. 84. Springer, Berlin (2006) [48] Liu, J., Jin, X., Tsui, K.C.: Autonomy Oriented Computing. Kluwer Academic Publishers, Bonston (2006) [49] Nedjah, N., Alba, E., de Macedo Mourelle, L. (eds.): Parallel Evolutionary Computations. Studies in Computational Intelligence, vol. 22. Springer, Berlin (2006) [50] Nedjah, N., de Macedo Mourelle, L. (eds.): Swarm Intelligent Systems. Studies in Computational Intelligence. Springer, Berlin (2006) [51] Pintér, J.D. (ed.): Global Optimization: Scientific and Engineering Case StudiesNonconvex Optimization and Its Applications, vol. 85. Springer, Berlin (2006) [52] Steyaert, M., van Roermund, A.H.M., Huijsing, J.H. (eds.): Analog Circuit Design. Springer, Berlin (2006) [53] Tiwari, A., Knowles, J., Avineri, E., Dahal, K., Roy, R. (eds.): Applications of Soft Computing: Recent Trends. Springer, Heidelberg (2006) [54] Wiak, S., Krawczyk, A., Trlep, M. (eds.): Computer Engineering in Applied Electromagnetism. Springer, Berlin (2006) [55] Zhang, H., Liu, D.: Fuzzy Modeling and Fuzzy Control. Birkhäuser, Boston (2006) [56] Zhang, G.Q., Van Driel, W.D., Fan, X.J. (eds.): Mechanics of Microelectronics. Springer, Berlin (2006) [57] Zomaya, A.Y. (ed.): Handbook of Nature-Inspired and Innovative Computing. Springer, Berlin (2006) [58] Zomaya, A.Y.: Parallel computing for bioinformatics and computational biology; models, enabling technologies, and case studies. John Wiley, New York (2006) [59] Chahl, J.S., Jain, L.C., Mizutani, A., Sato-Ilic, M. (eds.): Innovations in Intelligent Machines, vol. 1. Springer, Berlin (2007) [60] Cios, K.J., Pedrycz, W., Swiniarski, R.W., Kurgan, L.A.: Data Mining: A Knowledge Discovery Approach. Springer, Berlin (2007) [61] Corchado, E., Corchado, J.M., Abraham, A. (eds.): Innovations in Hybrid Intelligent Systems-Advances in Soft Computing, vol. 44. Springer, Berlin (2007) [62] Ebashi, S., Ohtsuki, I.: Regulatory Mechanisms of Striated Muscle ContractionAdvances in Experimental Medicine and Biology, vol. 592. Springer, Berlin (2007) [63] Grosan, C., Abraham, A., Ishibuchi, H. (eds.): Hybrid Evolutionary Algorithms. Studies in Computational Intelligence, vol. 75. Springer, Berlin (2007)

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[64] Jain, L.C., Palade, V., Srinivasan, D.: Advances in Evolutionary Computing for System Design. Studies in Computational Intelligence, vol. 66. Springer, Heidelberg (2007) [65] Kaburlasos, V.G., Ritter, G.X.: Computational Intelligence Based on Lattice Theory. Studies in Computational Intelligence, vol. 67. Springer, Berlin (2007) [66] Lobo, F.G., Lima, C.F., Michalewicz, Z. (eds.): Parameter Setting in Evolutionary Algorithms. Studies in Computational Intelligence, vol. 54. Springer, Berlin (2007) [67] Melin, P., Castillo, O., Ramírez, E.G., Kacprzyk, J., Pedrycz, W. (eds.): Analysis and Design of Intelligent Systems using Soft Computing Techniques-Advances in Soft Computing, vol. 41. Springer, Berlin (2007) [68] Nedjah, N., Abraham, A., de Macedo Mourelle, L. (eds.): Computational Intelligence in Information Assurance and Security. Studies in Computational Intelligence, vol. 57. Springer, Berlin (2007) [69] Nedjah, N., dos Santos Coelho, L., de Macedo Mourelle, L.: Mobile Robots: The Evolutionary Approach. Studies in Computational Intelligence, vol. 50. Springer, Berlin (2007) [70] Saad, A., Avineri, E., Dahal, K., Sarfraz, M., Roy, R.: Soft Computing in Industrial Applications: Recent and Emerging Methods and Techniques-Advances in Soft Computing, vol. 39. Springer, Berlin (2007) [71] Sobh, T., Elleithy, K., Mahmood, A., Karim, M.: Innovative Algorithms and Techniques in Automation, Industrial Electronics and Telecommunications. Springer, Berlin (2007) [72] Suri, J.S., Farag, A.A.: Deformable Models: Biomedical and Clinical Applications. Springer, Berlin (2007) [73] Törn, A., Žilinskas, J.: Models and Algorithms for Global Optimization: Essays Dedicated to Antanas Žilinskas on the Occasion of His 60th Birthday. Springer, Berlin (2007) [74] Valavanis, K.P. (ed.): Advances in Unmanned Aerial Vehicles: State of the Art and the Road to Autonomy. Springer, Berlin (2007) [75] Welcker, K.: Evolutionäre Algorithmen, Teubner (2007) [76] Yang, S., Ong, Y.S., Jin, Y.: Evolutionary Computation in Dynamic and Uncertain Environments. Studies in Computational Intelligence, vol. 51. Springer, Berlin (2007) [77] Abraham, A., Grosan, C., Pedrycz, W. (eds.): Engineering Evolutionary Intelligent Systems. Studies in Computational Intelligence, vol. 82. Springer, Berlin (2008) [78] Ao, S.I., Riger, B., Chen, S.S. (eds.): Advances in Computational Algorithms and Data Analysis. Lecture Notes Electrical Engineering, vol. 14. Springer, Berlin (2008) [79] Brabazon, A., O’Neill, M. (eds.): Natural Computing in Computational Finance. Studies in Computational Intelligence, vol. 100. Springer, Berlin (2008) [80] Castillo, O., Xu, L., Ao, S.I. (eds.): Trends in Intelligent Systems and Computer Engineering. Lecture Notes Electrical Engineering, vol. 6. Springer, Berlin (2008) [81] Chaturvedi, D.K.: Soft Computing: Techniques and Its Applications in Electrical Engineering. Studies in Computational Intelligence, vol. 103. Springer, Berlin (2008) [82] Cotta, C., Reich, S., Schaefer, R., Ligęza, A. (eds.): Knowledge-Driven Computing. Studies in Computational Intelligence, vol. 102. Springer, Berlin (2008) [83] Cotta, C., Seraux, M., Sörensen, K. (eds.): Adaptive and Multilevel MetaheuristicsStudies in Computational Intelligence, vol. 136. Springer, Heidelberg (2008)

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[84] Cotta, C., van Hemert, J. (eds.): Recent Advances in Evolutionary Computation for Combinatorial Optimization. Studies in Computational Intelligence, vol. 153. Springer, Berlin (2008) [85] Fulcher, J., Jain, L.C. (eds.): Computational Intelligence: a Compendium. Studies in Computational Intelligence, vol. 115. Springer, Berlin (2008) [86] Ghosh, A., Dehuri, S., Ghosh, S. (eds.): Multi-objective Evolutionary Algorithms for Knowledge Discovery from Databases. Studies in Computational Intelligence, vol. 98. Springer, Berlin (2008) [87] Grosse, C.U., Ohtsu, M. (eds.): Acoustic Emission Testing. Springer, Berlin (2008) [88] Kelemen, A., Abraham, A., Chen, Y. (eds.): Computational Intelligence in Bioinformatics. Studies in Computational Intelligence, vol. 94. Springer, Berlin (2008) [89] Kontoghiorghes, E.J., Rustem, B., Winker, P. (eds.): Computational Methods in Financial Engineering: Essays in Honour of Manfred Gilli. Springer, Berlin (2008) [90] Kramer, O.: Self-Adaptive Heuristics for Evolutionary Computation. Studies in Computational Intelligence, vol. 147. Springer, Berlin (2008) [91] Krasnogor, N., Nicosia, G., Pavone, M., Pelta, D. (eds.): Nature Inspired Cooperative Strategies for Optimization. Studies in Computational Intelligence, vol. 129. Springer, Berlin (2008) [92] Lee, K.Y., El-Sharkawi, M.A. (eds.): Modern Heuristic Optimization Techniques: Theory and Applications to Power Systems. Wiley-IEEE Press, New York (2008) [93] Liang, S. (ed.): Advances in Land Remote Sensing: System, Modeling, Inversion and Application. Springer, Berlin (2008) [94] Liu, Y., Sun, A., Loh, H.T., Lu, W.F., Lim, E.P. (eds.): Advances of Computational Intelligence in Industrial Systems. Studies in Computational Intelligence, vol. 116. Springer, Berlin (2008) [95] Prasad, B. (ed.): Soft Computing Applications in Industry. Studies in Fuzziness and Soft Computing, vol. 226. Springer, Berlin (2008) [96] Prasad, B. (ed.): Soft Computing Applications in Business. Studies in Fuzziness and Soft Computing, vol. 230. Springer, Berlin (2008) [97] Prokhorov, D. (ed.): Computational Intelligence in Automotive Applications. Studies in Computational Intelligence, vol. 132. Springer, Berlin (2008) [98] Riolo, R., Soule, T., Worzel, B. (eds.): Genetic Programming Theory and Practice, vol. 5. Springer, Berlin (2008) [99] Siarry, P., Michalewicz, Z. (eds.): Advances in Metaheuristics for Hard Optimization. Springer, Berlin (2008) [100] Smolinski, T.G., Milanova, M.G., Hassanien, A.E. (eds.): Applications of Computational Intelligence in Biology. Studies in Computational Intelligence, vol. 122. Springer, Berlin (2008) [101] Smolinski, T.G., Milanova, M.G.,, A.E.: Computational Intelligence in Biomedicine and Bioinformatics. Studies in Computational Intelligence, vol. 151. Springer, Berlin (2008) [102] Tizhoosh, H.R., Ventresca, M. (eds.): Oppositional Concepts in Computational Intelligence. Studies in Computational Intelligence, vol. 155. Springer, Berlin (2008) [103] Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems II. Springer, Berlin (2008) [104] Wiak, S., Krawczyk, A., Dolezel, I. (eds.): Intelligent Computer Techniques in Applied Electromagnetics. Studies in Computational Intelligence, vol. 119. Springer, Berlin (2008)

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[105] Xhafa, F., Abraham, A. (eds.): Metaheuristics for Scheduling in Industrial and Manufacturing Applications. Studies in Computational Intelligence, vol. 128. Springer, Berlin (2008) [106] Yang, A., Shan, Y., Bui, L.T.: Success in Evolutionary Computation. Studies in Computational Intelligence, vol. 92. Springer, Berlin (2008) [107] Zhang, J., Sanderson, A.C.: An approximate Gaussian model of differential evolution with spherical fitness function. In: 2007 IEEE Congress Evolutionary Computation, Singapore, september 25-28, pp. 2220–2228 (2007) [108] Montgomery, J.: Differential evolution: Difference vectors and movement in solution space. In: IEEE Congress Evolutionary Computation, Trondheim, Norway, May 18-21, pp. 2833–2840 (2009) [109] Lampinen, J., Zelinka, I.: On stagnation of the differential evolution algorithm. In: 6th Int. Mendel Conf. Soft Computing, Brno, Czech Republic, June 7-9, pp. 76–83 (2000) [110] Sukov, A., Borisov, A.: A study of search technique in differential evolution. In: 7th Int. MENDEL Conf. Soft Computing, Brno, Czech Republic, June 6-8, pp. 144–148 (2001) [111] Tomislav, Š.: Improving convergence properties of the differential evolution algorithm. In: 8th Int. MENDEL Conf. Soft Computing, Brno, Czech Republic, June 57, pp. 80–86 (2002) [112] Ali, M.M.: Differential evolution with preferential crossover. European J. Operational Research 181(3), 1137–1147 (2007) [113] Sutton, A.M., Lunacek, M., Whitley, L.D.: Differential evolution and nonseparability: using selective pressure to focus search. In: 2007 Genetic Evolutionary Computation Conf., London, UK, July 7-11, pp. 1428–1435 (2007) [114] Zaharie, D.: Statistical properties of differential evolution and related random search algorithms. In: 18th Symp. Computational Statistics, Oporto, Portugal, August 2429, pp. 473–485 (2008) [115] Dasguptu, S., Das, S., Biswas, A., Abraham, A.: On stability and convergence of the population-dynamics in differential evolution. AI Communications 22(1), 1–20 (2009) [116] Zielinski, K., Laur, R.: Variants of differential evolution for multi-objective optimization. In: 2007 IEEE Symp. Computational Intelligence Multicriteria Decision Making, Honolulu, HI, April 1-5, pp. 91–98 (2007) [117] Zaharie, D.: A comparative analysis of crossover variants in differential evolution. In: Int. Multiconference Computer Science Information Technology, pp. 171–181 (2007) [118] Zaharie, D.: Parameter adaption in differential evolution by controlling the population diversity. In: 4th Int. Workshop Symbolic Numeric Algorithms Scientific Computing, Timi¸ soara, Romania, October 9-12, pp. 385–397 (2002) [119] Zaharie, D.: Control of population diversity and adaptation in differential evolution algorithms. In: 9th Int. Mendel Conf. Soft Computing, Brno, Czech Republic, June 2003, pp. 41–46 (2003) [120] Hajji, O., Brisset, S., Brochet, P.: A stop criterion to accelerate magnetic optimization process using genetic algorithms and finite element analysis. IEEE Trans. Magnetics 39(3 I), 1297–1300 (2003)

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[121] Zielinski, K., Peters, D., Laur, R.: Run time analysis regarding stopping criteria for differential evolution and particle swarm optimization. In: 1st Int. Conf. Experiments/Process/System Modelling/Simulation/Optimization, Athens, Greece, July 6-9 (2005) [122] Zielinski, K., Peters, D., Laur, R.: Stopping criteria for single-objective optimization. In: 3rd Int. Conf. Computational Intelligence Robotics Autonomous Systems, Singapore, December 13-16 (2005) [123] Zielinski, K., Laur, R.: Stopping criteria for constrained optimization with particle swarms. In: 2nd Int. Conf. Bioinspired Optimization Methods Applications, Ljubljana, Slovenia, October 9-10, pp. 45–54 (2006) [124] Zielinski, K., Weitkemper, P., Laur, R., Kammeyer, K.D.: Examination of stopping criteria for differential evolution based on a power allocation problem. In: 10th Int. Conf. Optimization Electrical Electronic Equipment, Brasov, Romania, May 18-19, pp. 149–156 (2006) [125] Zielinski, K., Laur, R.: Stopping criteria for a constrained single-objective particle swarm optimization algorithm. Informatics (Ljubljana) 31(1), 51–59 (2007) [126] Wolpert, D.H., Macready, W.G.: No free lunch theorems for optimization. IEEE Trans. Evolutionary Computation 1(1), 67–82 (1997) [127] Michael, C., McGraw, G.: Opportunism and Diversity in Automated Software Test Data Generation, Technical Report RSTR-003-97-13, version 1.3, RST Corporation, Sterling, VA, USA (December 8, 1997) [128] Masters, T., Land, W.: A new training algorithm for the general regression neural network. In: 1997 IEEE Int. Conf. Systems Man Cybernetics, Orlando, FL, October 12-15, vol. 3, pp. 1990–1994 (1997) [129] Engle, R.F., Manganelli, S.: CAViaR: conditional autoregressive value at risk by regression quantiles, UCSD Economics Discussion Paper 99-20, University of California, San Diego, Department of Economics (October 1999) [130] Cafolla, A.A.: A new stochastic optimisation strategy for quantitative analysis of core level photoemission data. Surface Science 402-404, 561–565 (1998)

Chapter 2

Basics of Differential Evolution Anyong Qing

1

2.1 A Short History 2.1.1 Inception Differential evolution was proposed by K.V. Price and R. Storn in 1995 [1]. At that time, Price was asked to solve the Chebyshev polynomial fitting problem [1]-[5] by Storn [2], [5]. Initially, he tried to solve it by using genetic annealing algorithm [6]. However, although he eventually found the solution to the 5-dimensional Chebyshev polynomial fitting problem by using genetic annealing algorithm, he was frustrated to notice that genetic annealing algorithm fails to fulfill the three requirements for a practical optimization technique: strong global search capability, fast convergence, and user friendliness. A breakthrough happened when Price came up with an innovative scheme for generating trial parameter vectors. In this scheme, a new parameter vector is generated by adding the weighted difference vector between two population members to a third member. Such a scheme was named as differential mutation and has been well known to be the crucial idea behind the success of differential evolution. The cornerstone for differential evolution was therefore laid. Price wrapped up his invention with other critical ideas: natural real code, arithmetic operations, mother-child competition and selection, and execution of evolutionary operations in the order of mutation-crossover-selection. Consequently, differential evolution, a very reliable, efficient, robust, and simple evolutionary algorithm was developed. Anyong Qing Temasek Laboratories, National University of Singapore 5A, Engineering Dr 1 #06-09, Singapore 117411 e-mail: [email protected]

1

A. Qing and C.K. Lee: Differential Evolution in Electromagnetics, ALO 4, pp. 19–42. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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A. Qing

2.1.2 Early Years 2.1.2.1 Assessment Evaluation is one of the essential parts of algorithm development and has to be conducted right after an optimization algorithm is developed. It is the undeniable responsibility of the algorithm’s originator(s) to evaluate the proposed optimization algorithm. During the evaluation process, the originator(s) may gain more insights behind the concerned new optimization algorithm and make further efforts to improve it. The first evaluation of differential evolution was reported in the founding publication [1]. DE/rand/1/exp and DE/current-to-best/1/exp were evaluated over a test bed containing 7 unconstrained toy functions and 2 constrained toy functions. All intrinsic control parameters are fixed by trial and error approach. Comparison with annealed Nelder & Mead strategy and adaptive simulated annealing was also presented. Price and Storn, the originators, published two successive performance evaluation reports in 1996 [7] and 1997 [8]. Different strategies of differential evolution were evaluated over larger test beds. It is interesting to note that differential evolution quickly came to the attention of other researchers [9]. Evaluation of differential evolution over a test bed of 15 functions was carried out. Comparison with a variety of methods was also made. Differential evolution solves all functions successfully. It converges most rapidly while optimizing 11 of the 15 functions. 2.1.2.2 Reputation Building Differential evolution proved itself by winning in the two International Contests on Evolutionary Optimization (ICEO) [2], [3], [5], [10] in 1996 and 1997. More importantly, it was the best evolutionary algorithm among all entries since the first two places were won by non-evolutionary algorithms. It finished 3rd in the 1st International Contest on Evolutionary Optimization held in Nagoya, Japan from May 20, 1996 to May 22, 1996 [2], [5]. Differential evolution was the best among all qualified entries in the 2nd International Contests on Evolutionary Optimization [3] although the actual contest was cancelled due to lack of valid entries. 2.1.2.3 Applications Each and every practical optimization algorithm has to be picked up by people working in practical fields. More importantly, it will receive further evaluation through practical applications and gain more insights to improve it to better suit requirements arising from outstanding problems. There is no exception for differential evolution. It was soon applied by both the originators [11]-[15] and other application engineers [16]-[29] to solve various practical problems. Benefiting fields in these early applications of differential evolution are summarized in Table 2.1.

2 Basics of Differential Evolution

21

Table 2.1 Fields Benefiting from Early Application of Differential Evolution application

reference

Priority encoding transmission (PET)-redundancy assignment

[11]

Design of an IIR-filter

[12], [13]

Design of a howling remover

[14]

Redesign a switched capacitor (SC)-filter suffering from parasitic capacitances

[15]

multisensor fusion

[16], [19]-[21]

Parameter estimation of a batch bioprocess

[17]

Parallel optimization

[18]

Training of general regression neural network

[22]

Software test data generation

[23]

Scheduling of core blowers

[24]-[25]

Image registration

[27]

Temperature control of a chemical reactor system

[28]-[29]

Selection of control policy of a robotic arm

[29]

2.1.2.4 Promotion To boost the awareness of differential evolution and expand the community, Rainer established a website [5]. Latest contents include (a) (b) (c) (d) (e)

History, basics and practical advice Codes Demos Applications Useful relevant links

The website is still one of the prime resources for differential evolution. 2.1.2.5 Practical Advice Differential evolution involves intrinsic control parameters. It has been realized right from the beginning that differential evolution is dependent on proper setting of these intrinsic control parameters. To convenience differential evolution applicants, some practical advices to choose intrinsic control parameter values are recommended by the originators [3], [5], [8], [14], [30]. The recommendations are (a) Choose a population 2 [30], 5 [8], 10 [5], [14], [30] times problem dimension, or larger [3]. (b) Choose mutation intensity from [0.5, 1] [5], [14].

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(c) Choose crossover probability considerably lower than 1 if convergent, otherwise choose it from [0.8, 1] [5], [8], [14]. However, in [30], it is suggested to choose crossover probability “as large as possible without causing the population to become devoid of diversity”. (d) Choose lower mutation intensity for larger population size and vice versa [14]. (e) Adjust mutation intensity and crossover probability in parallel [3]. (f) Increase population size as crossover probability increases [3]. Some usage rules regarding generating initial population [14], formulating objective function [14], monitoring evolution [14], and strategy selection [3] are also recommended by the originators. 2.1.2.6 Standardization Strategies of differential evolution are denoted by DE/x/y/z where x indicates how the differential mutation base is chosen, y≥1 is the number of vector differences added to the base vector, and z is the law which the number of parameters donated by the mutant follows [3], [8]. The notation was inked by the originators in 1997 [8]. 2.1.2.7 More Adventures DE/rand/1/exp and DE/target-to-best/1/exp were proposed in the founding publication of differential evolution [1]. Later on, Price and Storn went on to explore better strategies. DE/best/2/bin [7], DE/random-to-best/1/exp [13]-[14] (corrected as DE/target-to-best/1/exp in [3]), DE/best/1/exp [14], DE/best/2/exp [10], [14], DE/current/1/exp [15], DE/rand/1/bin [8], [30], and other variants [2], were subsequently developed. It is very interesting to note that the inherent defect of differential evolutional, namely, slow convergence, was soon observed [17], [22]. Proposals to hybridize differential evolution with deterministic local optimizers were timely put forward.

2.1.3 Key Milestones in and after 1998 Differential evolution turned into fast track in 1998. Its potential has been increasingly realized by more and more researchers from ever growing number of fields. Researchers with different background have worked together to know more about its mathematical foundation, develop new strategies, or solve more challenging problems. Key events happening in and after 1998 in the short history of differential evolution are summarized in Table 2.2.

2 Basics of Differential Evolution

23

Table 2.2 Key Milestones in and after 1998 year

event

source

1998

first modification to initialization

[31]

1998

first modification to differential mutation

[31]

1998

first thesis on differential evolution

[32]

1999

first differential evolution for integer optimization parameters

[33]

1999

initial idea of dynamic differential evolution

[4], [34]-[35]

1999

Pareto differential evolution for multi-objective optimization

[36]-[37]

1999

first multi-population differential evolution

[38]

1999

first adaptive differential evolution

[39]

2000

first application in electromagnetics

[40]

2000

first empirical study on differential evolution

[41]-[42]

2000

first approximation of objective and constraint functions

[43]

2001

bibliography on differential evolution

[44]

2001

generalized differential evolution for multi-objective optimization

[45]

2005

first book on differential evolution

[3]

2005

non-dominated sorting differential evolution

[46]

2005

first strategy adaptation differential evolution

[47]

2006

first special session on differential evolution in IEEE Congress on Evolutionary Computation

2006

opposition-based differential evolution

[48]-[49]

2006

first system-level parametric study

[35]

2007

first thematic study on crossover in differential evolution

[50]

2009

evolutionary crimes

[4]

2009-

special issue on differential evolution in IEEE Transactions on Evolutionary

2010

Computation

2.2 The Foundational Differential Evolution Strategies Two differential evolution strategies, DE/rand/1/exp and DE/target-to-best/1/exp, were proposed in the founding publication of differential evolution [1] to minimize a single objective function f(x) with N-dimensional real optimization parameters x. Details of these two foundational strategies are given here to entertain readers with basic features of differential evolution.

2.2.1 Notations The notations applied in [4] are strictly followed here. However, for the convenience of readers and self-completeness of this book, primary notations are

24

A. Qing

summarized in Table 2.3. These notations will be consistently applied throughout this book unless specified otherwise. Secondary notations will be explained when it is mentioned for the first time. Table 2.3 Primary Notations notation f(x)

legend

notation

legend

x

N-dimensional vector of optimization parameters

objective function to minimize

xj

the jth optimization parameter

pc

crossover probability

b jL

lower bound of xj

bjU

upper bound of xj

P

population

Np

population size

P0

initial population

Pn

population of generation n

pi

the ith individual in P

pn,i

the ith individual in Pn

pbest

the best individual in P

pn,best

the best individual in Pn

pworst

the worst individual in P

pn,worst

the worst individual in Pn

x

i

i j

i

vector of optimization parameters of p i

x i

n,i

vector of optimization parameters of pn,i

n,i

the jth optimization parameter in xn,i of pn,i

x

the jth optimization parameter in x of p

xj

vi

mutant for pi

vn+1,i

mutant for pn,i

xv,i

vector of optimization parameters of vi

xn+1,v,i

vector of optimization parameters of vn+1,i

xj

v,i i

v,i

the jth optimization parameter in x of v

i

i

xj

n+1,v,i n,i

the jth optimization parameter in xn+1,v,i of vn+1,i base for vn+1,i

b

base for v

xb,i

vector of optimization parameters of bi

xn,b,i

vector of optimization parameters of bn,i

xjb,i

the jth optimization parameter in xb,i of bi

xjn,b,i

the jth optimization parameter in xn,b,i of bn,i

i

b

n+1,i

c

the ith child

c

xc,i

vector of optimization parameters of ci

xn+1,c,i

the ith child of generation n + 1 vector of optimization parameters of cn+1,i

xjc,i

the jth optimization parameter in xc,i of ci

xjn+1,c,i

the jth optimization parameter in xn+1,c,i of cn+1,i

F

mutation intensity

Fy

the yth mutation intensity

p1

index for donor 1 (one vector difference case)

p1y

index for donor 1 for the yth vector difference

p2

index for donor 2 (one vector difference case)

p2y

index for donor 2 for the yth vector difference

2.2.2 Strategy Framework Differential evolution optimizes f(x) with a population of Np individuals. It involves two stages, namely, initialization and evolution. Initialization generates initial population P0. Then the population evolves from one generation (Pn) to the next (Pn+1) until termination conditions are satisfied. While evolving from Pn to Pn+1, the three evolutionary operations, namely, differential mutation, crossover and selection, are executed in sequence. 2.2.2.1 Pseudo-code The Fortran-style pseudo-code of differential evolution is shown in Fig. 2.1.

2 Basics of Differential Evolution

25

Initialization n=0 do i = 1, Np generate p0,i evaluate f(x0,i) end do Evolution do while termination conditions are not satisfied n=n+1 do i = 1, Np differential mutation to obtain mutant vn+1,i crossover mutant vn+1,i with pn,i to deliver child cn+1,i evaluate child f(xn+1,c, i) selection to get individual pn+1,i end do end do

Fig. 2.1 Fortran-style Pseudo-code of Foundational Differential Evolution Strategies

2.2.2.2 Initialization Initialization generates the initial population P0 which contains Np individuals p0,i.

(

x 0j , i = b Lj + α ij bUj − b Lj

)

1≤ i ≤ Np, 1≤ j ≤ N

(1)

where the real random number, αji , is usually but not necessarily uniform in [0,1]. Alternatively, “in case a preliminary solution is available, the initial population is often generated by adding normally distributed random deviations to the nominal solution” x0 [1].

x

0, i j

⎧⎪ x 0j =⎨ 0 i ⎪⎩ x j + σ j

i =1 2 ≤ i ≤ N p −1

1≤ j ≤ N

(2)

where the real random number, σji , is usually but not necessarily “normally distributed”. 2.2.2.3 Differential Mutation Differential mutation generates a mutant vn+1,i for pn,i as follows

(

)

x n +1,v ,i = x n ,b ,i + F x n , p1 − x n , p2 , 1 ≤ i ≠ p1 ≠ p2 ≤ N p n,i

The differential mutation base b is chosen in two different ways in [1].

(3)

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A. Qing

A Random bn,i is randomly chosen from Pn and is different with both donors p n , p1 and p n , p2 . B Target-to-best bn,i is a point on a line between pn,i and pn,best. It is generated through the following arithmetic recombination

(

x n ,b ,i = x n ,i + λ x n ,best − x n ,i

)

(4)

where λ is the recombination coefficient. Alternatively, by looking at the general formulation of differential mutation shown in (2.2) in [1], this form of differential mutation, target-to-best/1, can be regarded as a biased form of current/2 in which the current individual pn,i serves as differential mutation base, the first vector difference is xn,best-xn,i weighted by λ, and the second vector difference is x n , p1 − x n , p2 weighted by F. 2.2.2.4 Crossover Crossover delivers a child cn+1,i through mating vn+1,i with pn,i. Exponential crossover is applied in [1]. The Fortran-style pseudo-code for exponential crossover is given in Fig. 2.2. A starting point r (1≤r≤N) is first randomly chosen. xrn+1,c,i of cn+1,i is taken from xrn+1,v,i of vn+1,i. Parameters of cn+1,i after (in cyclic sense) r depends on a series of Bernoulli experiments of probability pc, a constant in [0,1]. vn+1,i will keep donating its parameters to cn+1,i until the Bernoulli experiment is unsuccessful or the crossover length L, i.e., the number of parameters of the child donated by the mutant, is already N - 1. The remaining parameters of cn+1,i come from pn,i. do j = 1, N xjn+1,c,i = xjn,i end do r = N * rand(0, 1) + 1 k=r

xkn+1,c,i = xkn+1,v,i L=1

E = rand(0, 1) do while (E ” pc and L < N -1) L=L+1 k = 1 + mod(k, N) xkn+1,c,i = xkn+1,v,i

E = rand(0, 1) end do

Fig. 2.2 Fortran-style Pseudo-code of Exponential Crossover for Differential Evolution

2 Basics of Differential Evolution

27

A demonstrative example is shown in Fig. 2.3. N = 8, r = 7 and the crossover length is 3.

xn,i

28.69

35.09

57.82

12.06

26.99

82.96

65.30

52.68

xn+1,v,i

15.23

16.22

78.33

68.12

32.88

67.55

99.28

85.86

1

0

15.23

35.09

Bernoulli experiments

xn+1,c,i

1 57.82

12.06

26.99

82.96

99.28

85.86

Fig. 2.3 Exponential Crossover

2.2.2.5 Selection The selection operation follows Darwin’s natural selection, or survival of the fittest [51]. Child cn+1,i competes with a predetermined individual in population Pn and replaces it if cn+1,i dominates its competitor. cn+1,i usually competes with pn,i [12] although it is not stated explicitly in [1]. Such a selection operation is mathematically expressed as

p

n +1, i

⎧⎪c n +1,i f (x n +1,c, i ) ≤ f (x n, i ) = ⎨ n ,i ⎪⎩p otherwise

(5)

Sometimes, the selection is conducted in a sense of stronger dominance [10], i.e.,

⎧⎪c n +1, i f (x n +1,c, i ) < f (x n, i ) p n +1,i = ⎨ n ,i ⎪⎩p otherwise

(6)

As far as we know, by now, there is no thematic study on the effect of these two selection schemes. 2.2.2.6 Termination Conditions Termination conditions are not specified in [1]. However, other early publications by the originators [8]-[10] suggest that limit of number of generations is implemented to terminate the evolution process. It is yet to clarify whether “objective met” [3] is implemented although it is mentioned in [1] that the best individual pn+1,best in the new population Pn+1 is updated at the end of each evolution loop.

2.2.3 Intrinsic Control Parameters The two differential evolution strategies in [1] share three intrinsic control parameters

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A. Qing

(1) Population size Np (2) Mutation intensity F (3) Crossover probability pc In general, DE/target-to-best/1/exp has one more intrinsic control parameter, the recombination coefficient λ. However, for simplicity, λ is usually chosen identical with F.

2.3 Classic Differential Evolution Besides the aforementioned foundational differential evolution strategies proposed in the originators’ founding publication on differential evolution [1], many more differential evolution strategies under the same umbrella of classic differential evolution have been put forward. All classic differential evolution strategies share the same framework as shown in Fig. 2.1 while distinguish themselves in terms of initialization, differential mutation, crossover, objective function evaluation, selection, and termination conditions.

2.3.1 Initialization The two general initialization schemes have already been described by the originators clearly in their founding publication on differential evolution [1]. However, Different differential evolution strategy may apply different probability distribution i i function to generate random numbers αj in (1) [52] or σj in (2) [31].

2.3.2 Differential Mutation According to the standard notation DE/x/y/z, the general formulation of differential mutation for classic differential evolution is [1]

(

x n +1,v ,i = x n ,b ,i + ∑ Fy x y ≥1

n , p1 y

−x

n , p2 y

),

1 ≤ i ≠ p1 y ≠ p2 y ≤ N p

(7)

It is interesting to note that Storn [53] uses normalized vector differences to generate mutant vn+1,i which is mathematically expressed as

x n +1,v ,i = x n ,b ,i +

1 y

∑ F (x y

y ≥1

n , p1 y

−x

n , p2 y

),

1 ≤ i ≠ p1 y ≠ p2 y ≤ N p

(8)

Unless specified otherwise, we will stick to the general formulation shown in (7). Different differential mutation implements different differential mutation base n, p n, p bn,i and uses various number of vector differences x 1 y − x 2 y . Some of the most prominent differential mutation schemes are summarized here.

2 Basics of Differential Evolution

29

2.3.2.1 Current Individual pn,i serves as differential mutation base bn,i for vn+1,i. 2.3.2.2 Best Individual pn,best serves as differential mutation base bn,i for vn+1,i. 2.3.2.3 Better The differential mutation base bn,i for vn+1,i is randomly chosen from individuals dominating individual pn,i, i.e., d(bn,i, pn,i)=true where d(bn,i, pn,i) is the logic dominance function [4]. 2.3.2.4 Random The differential mutation base bn,i for vn+1,i is randomly chosen from Pn and is different with pn,i and all donors. 2.3.2.5 Mean The differential mutation base bn,i for vn+1,i is the geometrical center of Pn, i.e.,

x n ,b ,i =

Np

1 Np

∑x

n ,i

(9)

i =1

2.3.2.6 Best of Random The differential mutation base bn,i for vn+1,i is randomly chosen from Pn and dominates all donors which are also randomly chosen from Pn, i.e.,

(

d b n,i , p

n , p1 y

) = true

(

∩ d b n,i , p

n , p2 y

) = true

∀y

(10)

2.3.2.7 Arithmetic Best Arithmetic best is the synonym of target-to-best. 2.3.2.8 Arithmetic Better This scheme differs with arithmetic best by replacing pn,best in (4) with an individual dominating pn,i. A synonym for this scheme is target-to-better. 2.3.2.9 Arithmetic Random Likewise, in this scheme, pn,best in (4) is replaced by an individual randomly chosen from Pn. Its synonym is target-to-random.

30

A. Qing

2.3.2.10 Trigonometric Trigonometric differential mutation was proposed by Fan and Lampinen in 2003 [54]. Mutant vn+1,i is generated as

∑ ( f (x ) − 3

n , pk

x

n +1, v ,i

=

x

n , p1

+x

n , p2

+x

3

n , p3

+

j =1

(

)

( ))(x

f x

(

n, p j

n, p j

− x n , pk

(

)

)

f x n , p1 + f x n , p2 + f x n , p3

)

k = mod( j ,3) + 1 (11)

where individuals p n , p1 , p n , p2 , and p n , p3 may be chosen from Pn in different ways such as current, best, better, and random as aforementioned. Equivalently, the trigonometric differential mutation can be regarded as a special differential mutation in which the center of p n , p1 , p n , p2 , and p n , p3 acts as differential mutation base and p n , p1 , p n , p2 , and p n , p3 donates equally to vector differences. It can be seen that the intrinsic control parameter F does not show up in trigonometric differential mutation which might be one of its most attractive features. 2.3.2.11 Directed Directed differential mutation was also proposed by Fan and Lampinen in 2003 [55]. Mutant vn+1,i is generated as

( (

)( )

3 ⎡ f x n , p1 ⎤ n , p1 x n +1,v ,i = x n , p1 + ∑ ⎢1 − x − x n , pi n , pi ⎥ f x i =2 ⎣ ⎦

)

(12)

where individual p n , p1 dominates individuals p n , p2 and p n , p3 . It can be seen from (12) that individual p n , p1 serves as both differential mutation base and donor. Similarly, the intrinsic control parameter F does not show up.

2.3.3 Crossover Crossover has been thought unessential for differential evolution [4]. It is even not applied in some differential evolution strategies [3], [56]. However, recent studies hint that its significance in differential evolution might be seriously underestimated [4]. Crossover has been extensively studied in genetic algorithms. Almost all crossover schemes there can be implemented in differential evolution straightforward or after minor adjustment. Besides the exponential crossover mentioned earlier, some of the crossover schemes commonly applied in differential evolution are summarized here. In most evolutionary algorithms, the child cn+1,i is required to be different from its parents. This convention is followed here although it is not absolutely necessary in differential evolution.

2 Basics of Differential Evolution

31

2.3.3.1 Binary Crossover Binary crossover may be one of the most common crossover schemes in differential evolution. In this scheme, as shown in Fig. 2.4, cn+1,i inherits an optimization parameter from either vn+1,i or pn,i according to the result of a Bernoulli experiment of crossover probability pc, where β is a real random number uniform in [0,1]. do j = 1, N xjn+1,c,i = xjn,i end do L=0 do j = 1, N ȕ = rand(0, 1) if (ȕ ” pc) then L=L+1 xjn+1,c,i = xjn+1,v,i end if end do if (L = 0) then r = N * rand(0, 1) + 1 xrn+1,c,i = xrn+1,v,i else if (L = N) then r = N * rand(0, 1) + 1 xrn+1,c,i = xrn,i end if

Fig. 2.4 Fortran-style Pseudo-code of Binomial Crossover

A demonstrative example is shown in Fig. 2.5. cn+1,i inherits parameters x2n+1,v,i, x4 , x7n+1,v,i, and x8n+1,v,i from vn+1,i and x1n,i, x3n,i, x5n,i, x6n,i from pn,i. Therefore, the crossover length is 4. n+1,v,i

xn,i

28.69

35.09

57.82

12.06

26.99

82.96

65.30

52.68

xn+1,v,i

15.23

16.22

78.33

68.12

32.88

67.55

99.28

85.86

0

1

0

1

0

0

1

1

28.69

16.22

57.82

68.12

26.99

82.96

99.28

85.86

Bernoulli experiments

xn+1,c,i

Fig. 2.5 Binomial Crossover

32

A. Qing

2.3.3.2 One-Point Crossover One-point crossover randomly selects a single crossover point r (1