Asymptotic Cones and Functions in Optimization and Variational Inequalities / Edition 1

Asymptotic Cones and Functions in Optimization and Variational Inequalities / Edition 1

by Alfred Auslender, Marc Teboulle
     
 

ISBN-10: 0387955208

ISBN-13: 9780387955209

Pub. Date: 10/01/2002

Publisher: Springer New York

The book will serve as useful reference and self-contained text for researchers and graduate students in the fields of modern optimization theory and nonlinear analysis.  See more details below

Overview

The book will serve as useful reference and self-contained text for researchers and graduate students in the fields of modern optimization theory and nonlinear analysis.

Product Details

ISBN-13:
9780387955209
Publisher:
Springer New York
Publication date:
10/01/2002
Series:
Springer Monographs in Mathematics Series
Edition description:
2003
Pages:
249
Product dimensions:
9.21(w) x 6.14(h) x 0.63(d)

Table of Contents

Preface
1Convex Analysis and Set-Valued Maps: A Review1
1.1Convex Sets1
1.2Convex Functions9
1.3Support Functions17
1.4Set-Valued Maps20
2Asymptotic Cones and Functions25
2.1Definitions of Asymptotic Cones25
2.2Dual Characterization of Asymptotic Cones31
2.3Closedness Criteria32
2.4Continuous Convex Sets44
2.5Asymptotic Functions47
2.6Differential Calculus at Infinity60
2.7Application I: Semidefinite Optimization66
2.8Application II: Modeling and Smoothing Optimization Problems72
3Existence and Stability in Optimization Problems81
3.1Coercive Problems81
3.2Weak Coercivity85
3.3Asymptotically Level Stable Functions93
3.4Existence of Optimal Solutions96
3.5Stability for Constrained Problems100
3.6Dual Operations and Subdifferential Calculus107
3.7Additional Results in the Convex Case112
3.8The Feasibility Problem116
4Minimizing and Stationary Sequences119
4.1Optimality Conditions in Convex Minimization119
4.2Asymptotically Well-Behaved Functions124
4.3Error Bounds for Convex Inequality Systems133
4.4Stationary Sequences in Constrained Minimization140
5Duality in Optimization Problems145
5.1Perturbational-Conjugate Duality145
5.2Fenchel Duality154
5.3Lagrangian Duality157
5.4Zero Duality Gap for Special Convex Programs162
5.5Duality and Asymptotic Functions166
5.6Lagrangians and Minimax Theory170
5.7Duality and Stationary Sequences178
6Maximal Monotone Maps and Variational Inequalities183
6.1Maximal Monotone Maps183
6.2Minty Theorem186
6.3Convex Functionals and Maximal Monotonicity191
6.4Domains and Ranges of Maximal Monotone Maps195
6.5Asymptotic Functionals of Maximal Monotone Maps197
6.6Further Properties of Maximal Monotone Maps206
6.7Variational Inequalities Problems212
6.8Existence Results for Variational Inequalities214
6.9Duality for Variational Inequalities221
References233
Index of Notation243
Index245

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