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Convex Analysis and Nonlinear Optimization: Theory and Examples

Jonathan M. Borwein and Adrian S. Lewis
Publisher: 
Springer Verlag
Publication Date: 
2006
Number of Pages: 
310
Format: 
Hardcover
Edition: 
2
Series: 
CMS Books in Mathematics
Price: 
69.95
ISBN: 
0-387-29570-4
Category: 
Monograph
We do not plan to review this book.

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

xi

xii Contents

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

Dummy View - NOT TO BE DELETED