Optimization on Metric and Normed Spaces

1. Auflage 2010
ISBN: 978-0-387-88621-3
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 434 Seiten

"Optimization on Metric and Normed Spaces" is devoted to the recent progress in optimization on Banach spaces and complete metric spaces. Optimization problems are usually considered on metric spaces satisfying certain compactness assumptions which guarantee the existence of solutions and convergence of algorithms. This book considers spaces that do not satisfy such compactness assumptions. In order to overcome these difficulties, the book uses the Baire category approach and considers approximate solutions. Therefore, it presents a number of new results concerning penalty methods in constrained optimization, existence of solutions in parametric optimization, well-posedness of vector minimization problems, and many other results obtained in the last ten years. The book is intended for mathematicians interested in optimization and applied functional analysis.

Autoren/Hrsg.

Inhaltsverzeichnis

1;Preface;6
2;Contents;10
3;1 Introduction;16
3.1;1.1 Penalty methods;16
3.2;1.2 Generic existence of solutions of minimizationproblems;21
3.3;1.3 Comments;25
4;2 Exact Penalty in Constrained Optimization;26
4.1;2.1 Problems with a locally Lipschitzian constraint function;26
4.2;2.2 Proofs of Theorems 2.1–2.4;30
4.3;2.3 An optimization problem in a finite-dimensional space;35
4.4;2.4 Inequality-constrained problems with convex constraint functions;39
4.5;2.5 Proofs of Propositions 2.15, 2.17 and 2.18;43
4.6;2.6 Proof of Theorem 2.11;47
4.7;2.7 Optimization problems with mixed nonsmooth nonconvex constraints;51
4.8;2.8 Proof of Theorem 2.22;58
4.9;2.9 Optimization problems with smooth constraint and objective functions;63
4.10;2.10 Proofs of Theorems 2.26 and 2.27;67
4.11;2.11 Optimization problems in metric spaces;74
4.12;2.12 Proof of Theorem 2.35;79
4.13;2.13 An extension of Theorem 2.35;86
4.14;2.14 Exact penalty property and Mordukhovich basicsub differential;88
4.15;2.15 Proofs of Theorems 2.40 and 2.41;91
4.16;2.16 Comments;94
5;3 Stability of the Exact Penalty;95
5.1;3.1 Minimization problems with one constraint;95
5.2;3.2 Auxiliary results;100
5.3;3.3 Proof of Theorems 3.4 and 3.5;102
5.4;3.4 Problems with convex constraint functions;107
5.5;3.5 Proof of Theorem 3.12;112
5.6;3.6 An extension of Theorem 3.12 for problems with one constraint function;116
5.7;3.7 Proof of Theorem 3.14;118
5.8;3.8 Nonconvex inequality-constrained minimization problems;121
5.9;3.9 Proof of Theorem 3.16;126
5.10;3.10 Comments;134
6;4 Generic Well-Posedness of Minimization Problems;135
6.1;4.1 A generic variational principle;135
6.2;4.2 Two classes of minimization problems;137
6.3;4.3 The generic existence result for problem (P1);138
6.4;4.4 The weak topology on the space M;141
6.5;4.5 Proofs of Theorems 4.4 and 4.5;145
6.6;4.6 An extension of Theorem 4.4;147
6.7;4.7 The generic existence result for problem (P2);149
6.8;4.8 Proof of Theorem 4.19;152
6.9;4.9 A generic existence result in optimization;155
6.10;4.10 A basic lemma for Theorem 4.23;156
6.11;4.11 An auxiliary result;160
6.12;4.12 Proof of Theorem 4.23;161
6.13;4.13 Generic existence of solutions for problems with constraints;162
6.14;4.14 An auxiliary variational principle;162
6.15;4.15 The generic existence result for problem (P3);167
6.16;4.16 Proof of Theorem 4.31;168
6.17;4.17 The generic existence result for problem (P4);171
6.18;4.18 Proof of Theorem 4.33;173
6.19;4.19 Well-posedness of a class of minimization problems;175
6.20;4.20 Auxiliary results for Theorems 4.36 and 4.37;178
6.21;4.21 Auxiliary results for Theorem 4.38;180
6.22;4.22 Proofs of Theorems 4.36 and 4.37;182
6.23;4.23 Proof of Theorem 4.38;184
6.24;4.24 Generic well-posedness for a class of equilibrium problems;186
6.25;4.25 An auxiliary density result;188
6.26;4.26 A perturbation lemma;190
6.27;4.27 Proof of Theorem 4.48;192
6.28;4.28 Comment;194
7;5 Well-Posedness and Porosity;195
7.1;5.1 s-porous sets in a metric space;195
7.2;5.2 Well-posedness of optimization problems;197
7.3;5.3 A variational principle;200
7.4;5.4 Well-posedness and porosity for classes of minimization problems;204
7.5;5.5 Well-posedness and porosity in convex optimization;206
7.6;5.6 Proof of Theorem 5.10;208
7.7;5.7 A porosity result in convex minimization;214
7.8;5.8 Auxiliary results for Theorem 5.12;215
7.9;5.9 Proof of Theorem 5.12;217
7.10;5.10 A porosity result for variational problems arising in crystallography;219
7.11;5.11 The set M \ Mr is porous;221
7.12;5.12 Auxiliary results;222
7.13;5.13 Proof of Theorem 5.20;224
7.14;5.14 Porosity results for a class of equilibrium problems;230
7.15;5.15 The first porosity result;231
7.16;5.16 The second porosity result;233
7.17;5.17 The third porosity result;235
7.18;5.18 Comments;238
8;6 Parametric Optimization;239
8.1;6.1 Generic variational principle;239
8.2;6.2 Concretization of the hypothesis (H);241
8.3;6.3 Two generic existence results;246
8.4;6.4 A generic existence result in parametric optimization;251
8.5;6.5 Parametric optimization and porosity;252
8.6;6.6 A variational principle and porosity;253
8.7;6.7 Concretization of the variational principle;258
8.8;6.8 Existence results for the problem (P2);262
8.9;6.9 Existence results for the problem (P1);269
8.10;6.10 Parametric optimization problems with constraints;271
8.11;6.11 Proof of Theorem 6.25;273
8.12;6.12 Comments;279
9;7 Optimization with Increasing Objective Functions;280
9.1;7.1 Preliminaries;280
9.2;7.2 A variational principle;281
9.3;7.3 Spaces of increasing coercive functions;286
9.4;7.4 Proof of Theorem 7.4;287
9.5;7.5 Spaces of increasing noncoercive functions;290
9.6;7.6 Proof of Theorem 7.10;291
9.7;7.7 Spaces of increasing quasiconvex functions;293
9.8;7.8 Proof of Theorem 7.14;294
9.9;7.9 Spaces of increasing convex functions;299
9.10;7.10 Proof of Theorem 7.21;301
9.11;7.11 The generic existence result for the minimization problem (P2);302
9.12;7.12 Proofs of Theorems 7.29 and 7.30;304
9.13;7.13 Well-posedness of optimization problems with increasing cost functions;308
9.14;7.14 Variational principles;311
9.15;7.15 Spaces of increasing functions;316
9.16;7.16 Comments;322
10;8 Generic Well-Posedness of Minimization Problems with Constraints;323
10.1;8.1 Variational principles;323
10.2;8.2 Proof of Proposition 8.2;325
10.3;8.3 Minimization problems with mixed continuous constraints;329
10.4;8.4 An auxiliary result for (A2);332
10.5;8.5 An auxiliary result for (A3);333
10.6;8.6 An auxiliary result for (A4);334
10.7;8.7 Proof of Theorems 8.4 and 8.5;339
10.8;8.8 An abstract implicit function theorem;340
10.9;8.9 Proof of Theorem 8.10;341
10.10;8.10 An extension of the classical implicit function;345
10.11;8.11 Minimization problems with mixed smooth constraints;348
10.12;8.12 Auxiliary results;350
10.13;8.13 An auxiliary result for hypothesis (A4);353
10.14;8.14 Proof of Theorems 8.15 and 8.16;358
10.15;8.15 Comments;359
11;9 Vector Optimization;360
11.1;9.1 Generic and density results in vector optimization;360
11.2;9.2 Proof of Proposition 9.1;361
11.3;9.3 Auxiliary results;363
11.4;9.4 Proof of Theorem 9.2;371
11.5;9.5 Proof of Theorem 9.3;372
11.6;9.6 Vector optimization with continuous objective functions;379
11.7;9.7 Preliminaries;381
11.8;9.8 Auxiliary results;382
11.9;9.9 Proof of Theorem 9.9;388
11.10;9.10 Vector optimization with semicontinuous objective functions;392
11.11;9.11 Auxiliary results for Theorem 9.14;395
11.12;9.12 Proof of Theorem 9.14;399
11.13;9.13 Density results;401
11.14;9.14 Comments;405
12;10 Infinite Horizon Problems;406
12.1;10.1 Minimal solutions for discrete-time control systems in metric spaces;406
12.2;10.2 Auxiliary results;408
12.3;10.3 Proof of Theorem 10.2;412
12.4;10.4 Properties of good sequences;418
12.5;10.5 Convex discrete-time control systems in a Banach space;419
12.6;10.6 Preliminary results;421
12.7;10.7 Proofs of Theorems 10.13 and 10.14;424
12.8;10.8 Control systems on metric spaces;431
12.9;10.9 Proof of Proposition 10.23;432
12.10;10.10 An auxiliary result for Theorem 10.24;434
12.11;10.11 Proof of Theorem 10.24;435
12.12;10.12 Comments;436
13;References;437
14;Index;442