Contents
0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Background 7
1.1 Euclidean spaces . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Symmetric matrices . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Inequality constraints 22
2.1 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Theorems of the alternative . . . . . . . . . . . . . . . . . . . 30
2.3 Max-functions and first order conditions . . . . . . . . . . . . 36
3 Fenchel duality 42
3.1 Subgradients and convex functions . . . . . . . . . . . . . . . 42
3.2 The value function . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 The Fenchel conjugate . . . . . . . . . . . . . . . . . . . . . . 61
4 Convex analysis 78
4.1 Continuity of convex functions . . . . . . . . . . . . . . . . . . 78
4.2 Fenchel biconjugation . . . . . . . . . . . . . . . . . . . . . . . 90
4.3 Lagrangian duality . . . . . . . . . . . . . . . . . . . . . . . . 103
5 Special cases 113
5.1 Polyhedral convex sets and functions . . . . . . . . . . . . . . 113
5.2 Functions of eigenvalues . . . . . . . . . . . . . . . . . . . . . 120
5.3 Duality for linear and semidefinite programming . . . . . . . . 126
5.4 Convex process duality . . . . . . . . . . . . . . . . . . . . . . 132
6 Nonsmooth optimization 143
6.1 Generalized derivatives . . . . . . . . . . . . . . . . . . . . . . 143
3
6.2 Nonsmooth regularity and strict differentiability . . . . . . . . 151
6.3 Tangent cones . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.4 The limiting subdifferential . . . . . . . . . . . . . . . . . . . 167
7 The Karush-Kuhn-Tucker theorem 176
7.1 An introduction to metric regularity . . . . . . . . . . . . . . 176
7.2 The Karush-Kuhn-Tucker theorem . . . . . . . . . . . . . . . 184
7.3 Metric regularity and the limiting subdifferential . . . . . . . . 191
7.4 Second order conditions . . . . . . . . . . . . . . . . . . . . . 197
8 Fixed points 204
8.1 Brouwer’s fixed point theorem . . . . . . . . . . . . . . . . . . 204
8.2 Selection results and the Kakutani-Fan fixed point theorem . . 216
8.3 Variational inequalities . . . . . . . . . . . . . . . . . . . . . . 227
9 Postscript: infinite versus finite dimensions 238
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
9.2 Finite dimensionality . . . . . . . . . . . . . . . . . . . . . . . 240
9.3 Counterexamples and exercises . . . . . . . . . . . . . . . . . . 243
9.4 Notes on previous chapters . . . . . . . . . . . . . . . . . . . . 249
9.4.1 Chapter 1: Background . . . . . . . . . . . . . . . . . . 249
9.4.2 Chapter 2: Inequality constraints . . . . . . . . . . . . 249
9.4.3 Chapter 3: Fenchel duality . . . . . . . . . . . . . . . . 249
9.4.4 Chapter 4: Convex analysis . . . . . . . . . . . . . . . 250
9.4.5 Chapter 5: Special cases . . . . . . . . . . . . . . . . . 250
9.4.6 Chapter 6: Nonsmooth optimization . . . . . . . . . . 250
9.4.7 Chapter 7: The Karush-Kuhn-Tucker theorem . . . . . 251
9.4.8 Chapter 8: Fixed points . . . . . . . . . . . . . . . . . 251
10 List of results and notation 252
10.1 Named results and exercises . . . . . . . . . . . . . . . . . . . 252
10.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Bibliography 276
Index

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