Convex analysis in general vector spaces 9812380671, 9789812380678, 9789812777096
The primary aim of this book is to present the conjugate and subdifferential calculus using the method of perturbation f
319 64 4MB
English Pages 387 Year 2002
Table of contents :
Cover......Page 1
Title......Page 2
Title page......Page 3
Date-line......Page 4
Dedication......Page 5
Preface......Page 7
Contents......Page 9
Introduction......Page 11
1.1 Preliminary notions and results......Page 21
1.2 Closedness and interiority notions......Page 29
1.3 Open mapping theorems......Page 39
1.4 Variational principles......Page 49
1.5 Exercises......Page 54
1.6 Bibliographical notes......Page 56
2.1 Convex functions......Page 59
2.2 Semi-continuity of convex functions......Page 80
2.3 Conjugate functions......Page 95
2.4 The subdifferential of a convex function......Page 99
2.5 The general problem of convex programming......Page 119
2.6 Perturbed problems......Page 126
2.7 The fundamental duality formula......Page 133
2.8 Formulas for conjugates and $\varepsilon$-subdifferentials, duality relations and optimality conditions......Page 141
2.9 Convex optimization with constraints......Page 156
2.10 A minimax theorem......Page 163
2.11 Exercises......Page 166
2.12 Bibliographical notes......Page 175
3.1 Further fundamental results in convex analysis......Page 179
3.2 Convexity and monotonicity of subdifferentials......Page 189
3.3 Some classes of functions of a real variable and differentiability of convex functions......Page 208
3.4 Well conditioned functions......Page 215
3.5 Uniformly convex and uniformly smooth convex functions......Page 223
3.6 Uniformly convex and uniformly smooth convex functions on bounded sets......Page 241
3.7 Applications to the geometry of normed spaces......Page 246
3.8 Applications to the best approximation problem......Page 257
3.9 Characterizations of convexity in terms of smoothness......Page 263
3.10 Weak sharp minima, well-behaved functions and global error bounds for convex inequalities......Page 268
3.11 Monotone multifunctions......Page 289
3.12 Exercises......Page 308
3.13 Bibliographical notes......Page 312
Exercises - Solutions......Page 317
Bibliography......Page 369
Index......Page 379
Symbols and Notations......Page 383
Cover......Page 1
Title......Page 2
Title page......Page 3
Date-line......Page 4
Dedication......Page 5
Preface......Page 7
Contents......Page 9
Introduction......Page 11
1.1 Preliminary notions and results......Page 21
1.2 Closedness and interiority notions......Page 29
1.3 Open mapping theorems......Page 39
1.4 Variational principles......Page 49
1.5 Exercises......Page 54
1.6 Bibliographical notes......Page 56
2.1 Convex functions......Page 59
2.2 Semi-continuity of convex functions......Page 80
2.3 Conjugate functions......Page 95
2.4 The subdifferential of a convex function......Page 99
2.5 The general problem of convex programming......Page 119
2.6 Perturbed problems......Page 126
2.7 The fundamental duality formula......Page 133
2.8 Formulas for conjugates and $\varepsilon$-subdifferentials, duality relations and optimality conditions......Page 141
2.9 Convex optimization with constraints......Page 156
2.10 A minimax theorem......Page 163
2.11 Exercises......Page 166
2.12 Bibliographical notes......Page 175
3.1 Further fundamental results in convex analysis......Page 179
3.2 Convexity and monotonicity of subdifferentials......Page 189
3.3 Some classes of functions of a real variable and differentiability of convex functions......Page 208
3.4 Well conditioned functions......Page 215
3.5 Uniformly convex and uniformly smooth convex functions......Page 223
3.6 Uniformly convex and uniformly smooth convex functions on bounded sets......Page 241
3.7 Applications to the geometry of normed spaces......Page 246
3.8 Applications to the best approximation problem......Page 257
3.9 Characterizations of convexity in terms of smoothness......Page 263
3.10 Weak sharp minima, well-behaved functions and global error bounds for convex inequalities......Page 268
3.11 Monotone multifunctions......Page 289
3.12 Exercises......Page 308
3.13 Bibliographical notes......Page 312
Exercises - Solutions......Page 317
Bibliography......Page 369
Index......Page 379
Symbols and Notations......Page 383
