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

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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.
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Editorial Reviews

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"The main purpose of this book is to provide a systematic study of asymptotic cones and asymptotic functions in finite dimensional normed spaces. … Every chapter ends with bibliographical notes. … The book is addressed to graduate students at an advanced level and to researchers and practitioners in the fields of optimization theory, nonlinear programming and applied mathematical sciences. ... We recommend this book to all those who are interested in asymptotic analysis and its use." (Constantin Zalinescu, Zentralblatt MATH, Vol. 1017, 2003)

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Product Details

  • ISBN-13: 9780387955209
  • Publisher: Springer New York
  • Publication date: 10/1/2002
  • Series: Springer Monographs in Mathematics Series
  • Edition description: 2003
  • Edition number: 1
  • Pages: 249
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.63 (d)

Table of Contents

1 Convex Analysis and Set-Valued Maps: A Review 1
1.1 Convex Sets 1
1.2 Convex Functions 9
1.3 Support Functions 17
1.4 Set-Valued Maps 20
2 Asymptotic Cones and Functions 25
2.1 Definitions of Asymptotic Cones 25
2.2 Dual Characterization of Asymptotic Cones 31
2.3 Closedness Criteria 32
2.4 Continuous Convex Sets 44
2.5 Asymptotic Functions 47
2.6 Differential Calculus at Infinity 60
2.7 Application I: Semidefinite Optimization 66
2.8 Application II: Modeling and Smoothing Optimization Problems 72
3 Existence and Stability in Optimization Problems 81
3.1 Coercive Problems 81
3.2 Weak Coercivity 85
3.3 Asymptotically Level Stable Functions 93
3.4 Existence of Optimal Solutions 96
3.5 Stability for Constrained Problems 100
3.6 Dual Operations and Subdifferential Calculus 107
3.7 Additional Results in the Convex Case 112
3.8 The Feasibility Problem 116
4 Minimizing and Stationary Sequences 119
4.1 Optimality Conditions in Convex Minimization 119
4.2 Asymptotically Well-Behaved Functions 124
4.3 Error Bounds for Convex Inequality Systems 133
4.4 Stationary Sequences in Constrained Minimization 140
5 Duality in Optimization Problems 145
5.1 Perturbational-Conjugate Duality 145
5.2 Fenchel Duality 154
5.3 Lagrangian Duality 157
5.4 Zero Duality Gap for Special Convex Programs 162
5.5 Duality and Asymptotic Functions 166
5.6 Lagrangians and Minimax Theory 170
5.7 Duality and Stationary Sequences 178
6 Maximal Monotone Maps and Variational Inequalities 183
6.1 Maximal Monotone Maps 183
6.2 Minty Theorem 186
6.3 Convex Functionals and Maximal Monotonicity 191
6.4 Domains and Ranges of Maximal Monotone Maps 195
6.5 Asymptotic Functionals of Maximal Monotone Maps 197
6.6 Further Properties of Maximal Monotone Maps 206
6.7 Variational Inequalities Problems 212
6.8 Existence Results for Variational Inequalities 214
6.9 Duality for Variational Inequalities 221
References 233
Index of Notation 243
Index 245
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