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http://www.amazon.com/Convex-Analysis-Nonlinear-Optimization-Mathematics/dp/0387989404
Convex Analysis and Nonlinear Optimization
Theory and Examples
Series: CMS Books in Mathematics
Borwein, Jonathan, Lewis, Adrian S.
2nd ed., 2006, XII, 310 p., Hardcover
Optimization is a rich and thriving mathematical discipline. The theory underlying current computational optimization techniques grows ever more sophisticated. The powerful and elegant language of convex analysis unifies much of this theory. The aim of this book is to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. It can serve as a teaching text, at roughly the level of first year graduate students. While the main body of the text is self-contained, each section concludes with an often extensive set of optional exercises. The new edition adds material on semismooth optimization, as well as several new proofs that will make this book even more self-contained.
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Contents
Preface vii
1 Background 1
1.1 Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 SymmetricMatrices . . . . . . . . . . . . . . . . . . . . . . 9
2 Inequality Constraints 15
2.1 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . 15
2.2 Theorems of the Alternative . . . . . . . . . . . . . . . . . . 23
2.3 Max-functions . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Fenchel Duality 33
3.1 Subgradients and Convex Functions . . . . . . . . . . . . . 33
3.2 The Value Function . . . . . . . . . . . . . . . . . . . . . . 43
3.3 The Fenchel Conjugate . . . . . . . . . . . . . . . . . . . . . 49
4 ConvexAnalysis 65
4.1 Continuity of Convex Functions . . . . . . . . . . . . . . . . 65
4.2 Fenchel Biconjugation . . . . . . . . . . . . . . . . . . . . . 76
4.3 Lagrangian Duality . . . . . . . . . . . . . . . . . . . . . . . 88
5 Special Cases 97
5.1 Polyhedral Convex Sets and Functions . . . . . . . . . . . . 97
5.2 Functions of Eigenvalues . . . . . . . . . . . . . . . . . . . . 104
5.3 Duality for Linear and Semidefinite Programming . . . . . . 109
5.4 Convex Process Duality . . . . . . . . . . . . . . . . . . . . 114
6 Nonsmooth Optimization 123
6.1 Generalized Derivatives . . . . . . . . . . . . . . . . . . . . 123
6.2 Regularity and Strict Differentiability . . . . . . . . . . . . 130
6.3 Tangent Cones . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.4 The Limiting Subdifferential . . . . . . . . . . . . . . . . . . 145
7 Karush–Kuhn–Tucker Theory 153
7.1 An Introduction to Metric Regularity . . . . . . . . . . . . 153
7.2 The Karush–Kuhn–Tucker Theorem . . . . . . . . . . . . . 160
7.3 Metric Regularity and the Limiting Subdifferential . . . . . 166
7.4 Second Order Conditions . . . . . . . . . . . . . . . . . . . 172
8 Fixed Points 179
8.1 The Brouwer Fixed Point Theorem . . . . . . . . . . . . . . 179
8.2 Selection and the Kakutani–Fan Fixed Point Theorem . . . 190
8.3 Variational Inequalities . . . . . . . . . . . . . . . . . . . . . 200
9 More Nonsmooth Structure 213
9.1 Rademacher’s Theorem . . . . . . . . . . . . . . . . . . . . 213
9.2 Proximal Normals and Chebyshev Sets . . . . . . . . . . . . 218
9.3 Amenable Sets and Prox-Regularity . . . . . . . . . . . . . 228
9.4 Partly Smooth Sets . . . . . . . . . . . . . . . . . . . . . . . 233
10 Postscript: Infinite Versus Finite Dimensions 239
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
10.2 Finite Dimensionality . . . . . . . . . . . . . . . . . . . . . 241
10.3 Counterexamples and Exercises . . . . . . . . . . . . . . . . 244
10.4 Notes on Previous Chapters . . . . . . . . . . . . . . . . . . 248
11 List of Results and Notation 253
11.1 Named Results . . . . . . . . . . . . . . . . . . . . . . . . . 253
11.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Bibliography 275
Index 289
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