Optimization on Metric and Normed Spaces (eBook)

Preface 6
Contents 10
1 Introduction 16
1.1 Penalty methods 16
1.2 Generic existence of solutions of minimizationproblems 21
1.3 Comments 25
2 Exact Penalty in Constrained Optimization 26
2.1 Problems with a locally Lipschitzian constraint function 26
2.2 Proofs of Theorems 2.1–2.4 30
2.3 An optimization problem in a finite-dimensional space 35
2.4 Inequality-constrained problems with convex constraint functions 39
2.5 Proofs of Propositions 2.15, 2.17 and 2.18 43
2.6 Proof of Theorem 2.11 47
2.7 Optimization problems with mixed nonsmooth nonconvex constraints 51
2.8 Proof of Theorem 2.22 58
2.9 Optimization problems with smooth constraint and objective functions 63
2.10 Proofs of Theorems 2.26 and 2.27 67
2.11 Optimization problems in metric spaces 74
2.12 Proof of Theorem 2.35 79
2.13 An extension of Theorem 2.35 86
2.14 Exact penalty property and Mordukhovich basicsub differential 88
2.15 Proofs of Theorems 2.40 and 2.41 91
2.16 Comments 94
3 Stability of the Exact Penalty 95
3.1 Minimization problems with one constraint 95
3.2 Auxiliary results 100
3.3 Proof of Theorems 3.4 and 3.5 102
3.4 Problems with convex constraint functions 107
3.5 Proof of Theorem 3.12 112
3.6 An extension of Theorem 3.12 for problems with one constraint function 116
3.7 Proof of Theorem 3.14 118
3.8 Nonconvex inequality-constrained minimization problems 121
3.9 Proof of Theorem 3.16 126
3.10 Comments 134
4 Generic Well-Posedness of Minimization Problems 135
4.1 A generic variational principle 135
4.2 Two classes of minimization problems 137
4.3 The generic existence result for problem (P1) 138
4.4 The weak topology on the space M 141
4.5 Proofs of Theorems 4.4 and 4.5 145
4.6 An extension of Theorem 4.4 147
4.7 The generic existence result for problem (P2) 149
4.8 Proof of Theorem 4.19 152
4.9 A generic existence result in optimization 155
4.10 A basic lemma for Theorem 4.23 156
4.11 An auxiliary result 160
4.12 Proof of Theorem 4.23 161
4.13 Generic existence of solutions for problems with constraints 162
4.14 An auxiliary variational principle 162
4.15 The generic existence result for problem (P3) 167
4.16 Proof of Theorem 4.31 168
4.17 The generic existence result for problem (P4) 171
4.18 Proof of Theorem 4.33 173
4.19 Well-posedness of a class of minimization problems 175
4.20 Auxiliary results for Theorems 4.36 and 4.37 178
4.21 Auxiliary results for Theorem 4.38 180
4.22 Proofs of Theorems 4.36 and 4.37 182
4.23 Proof of Theorem 4.38 184
4.24 Generic well-posedness for a class of equilibrium problems 186
4.25 An auxiliary density result 188
4.26 A perturbation lemma 190
4.27 Proof of Theorem 4.48 192
4.28 Comment 194
5 Well-Posedness and Porosity 195
5.1 s-porous sets in a metric space 195
5.2 Well-posedness of optimization problems 197
5.3 A variational principle 200
5.4 Well-posedness and porosity for classes of minimization problems 204
5.5 Well-posedness and porosity in convex optimization 206
5.6 Proof of Theorem 5.10 208
5.7 A porosity result in convex minimization 214
5.8 Auxiliary results for Theorem 5.12 215
5.9 Proof of Theorem 5.12 217
5.10 A porosity result for variational problems arising in crystallography 219
5.11 The set M / Mr is porous 221
5.12 Auxiliary results 222
5.13 Proof of Theorem 5.20 224
5.14 Porosity results for a class of equilibrium problems 230
5.15 The first porosity result 231
5.16 The second porosity result 233
5.17 The third porosity result 235
5.18 Comments 238
6 Parametric Optimization 239
6.1 Generic variational principle 239
6.2 Concretization of the hypothesis (H) 241
6.3 Two generic existence results 246
6.4 A generic existence result in parametric optimization 251
6.5 Parametric optimization and porosity 252
6.6 A variational principle and porosity 253
6.7 Concretization of the variational principle 258
6.8 Existence results for the problem (P2) 262
6.9 Existence results for the problem (P1) 269
6.10 Parametric optimization problems with constraints 271
6.11 Proof of Theorem 6.25 273
6.12 Comments 279
7 Optimization with Increasing Objective Functions 280
7.1 Preliminaries 280
7.2 A variational principle 281
7.3 Spaces of increasing coercive functions 286
7.4 Proof of Theorem 7.4 287
7.5 Spaces of increasing noncoercive functions 290
7.6 Proof of Theorem 7.10 291
7.7 Spaces of increasing quasiconvex functions 293
7.8 Proof of Theorem 7.14 294
7.9 Spaces of increasing convex functions 299
7.10 Proof of Theorem 7.21 301
7.11 The generic existence result for the minimization problem (P2) 302
7.12 Proofs of Theorems 7.29 and 7.30 304
7.13 Well-posedness of optimization problems with increasing cost functions 308
7.14 Variational principles 311
7.15 Spaces of increasing functions 316
7.16 Comments 322
8 Generic Well-Posedness of Minimization Problems with Constraints 323
8.1 Variational principles 323
8.2 Proof of Proposition 8.2 325
8.3 Minimization problems with mixed continuous constraints 329
8.4 An auxiliary result for (A2) 332
8.5 An auxiliary result for (A3) 333
8.6 An auxiliary result for (A4) 334
8.7 Proof of Theorems 8.4 and 8.5 339
8.8 An abstract implicit function theorem 340
8.9 Proof of Theorem 8.10 341
8.10 An extension of the classical implicit function 345
8.11 Minimization problems with mixed smooth constraints 348
8.12 Auxiliary results 350
8.13 An auxiliary result for hypothesis (A4) 353
8.14 Proof of Theorems 8.15 and 8.16 358
8.15 Comments 359
9 Vector Optimization 360
9.1 Generic and density results in vector optimization 360
9.2 Proof of Proposition 9.1 361
9.3 Auxiliary results 363
9.4 Proof of Theorem 9.2 371
9.5 Proof of Theorem 9.3 372
9.6 Vector optimization with continuous objective functions 379
9.7 Preliminaries 381
9.8 Auxiliary results 382
9.9 Proof of Theorem 9.9 388
9.10 Vector optimization with semicontinuous objective functions 392
9.11 Auxiliary results for Theorem 9.14 395
9.12 Proof of Theorem 9.14 399
9.13 Density results 401
9.14 Comments 405
10 Infinite Horizon Problems 406
10.1 Minimal solutions for discrete-time control systems in metric spaces 406
10.2 Auxiliary results 408
10.3 Proof of Theorem 10.2 412
10.4 Properties of good sequences 418
10.5 Convex discrete-time control systems in a Banach space 419
10.6 Preliminary results 421
10.7 Proofs of Theorems 10.13 and 10.14 424
10.8 Control systems on metric spaces 431
10.9 Proof of Proposition 10.23 432
10.10 An auxiliary result for Theorem 10.24 434
10.11 Proof of Theorem 10.24 435
10.12 Comments 436
References 437
Index 442