Table of contents for Convex analysis and nonlinear optimization : theory and examples / Jonathan M. Borwein, Adrian S. Lewis.


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1 Background                                                 1
1.1  Euclidean  Spaces  ...................      . . . .  .  1
1.2  Symmetric Matrices  . .................... .         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 Convex Analysis                                           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



Library of Congress subject headings for this publication: Convex functions, Mathematical optimization, Nonlinear theories