Singer Duality for Nonconvex Approximation and Optimization

1. Auflage 2007
ISBN: 978-0-387-28395-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 356 Seiten

Reihe: CMS Books in Mathematics

The theory of convex optimization has been constantly developing over the past 30 years.  Most recently, many researchers have been studying more complicated classes of problems that still can be studied by means of convex analysis, so-called 'anticonvex' and 'convex-anticonvex' optimizaton problems.  This manuscript contains an exhaustive presentation of the duality for these classes of problems and some of its generalization in the framework of abstract convexity.  This manuscript will be of great interest for experts in this and related fields.

Autoren/Hrsg.

Singer, Ivan

Inhaltsverzeichnis

1;List of Figures;10
2;Contents;7
3;Preface;11
4;Preliminaries;18
4.1;1.1 Some preliminaries from convex analysis;18
4.2;1.2 Some preliminaries from abstract convex analysis;44
4.3;1.3 Duality for best approximation by elements of convex sets;56
4.4;1.4 Duality for convex and quasi-convex infimization;63
5;Worst Approximation;102
5.1;2.1 The deviation of a set from an element;103
5.2;2.2 Characterizations and existence of farthest points;110
6;Duality for Quasi- convex Supremization;118
6.1;3.1 Some hyperplane theorems of surrogate duality;120
6.2;3.2 Unconstrained surrogate dual problems for quasi- convex supremization;125
6.3;3.3 Constrained surrogate dual problems for quasi- convex supremization;138
6.4;3.4 Lagrangian duality for convex supremization;144
6.5;3.5 Duality for quasi-convex supremization over structured primal constraint sets;148
7;Optimal Solutions for Quasi- convex Maximization;153
7.1;4.1 Maximum points of quasi- convex functions;153
7.2;4.2 Maximum points of continuous convex functions;160
7.3;4.3 Some basic subdifferential characterizations of maximum points;165
8;Reverse Convex Best Approximation;169
8.1;5.1 The distance to the complement of a convex set;170
8.2;5.2 Characterizations and existence of elements of best approximation in complements of convex sets;177
9;Unperturbational Duality for Reverse Convex Infimization;184
9.1;6.1 Some hyperplane theorems of surrogate duaUty;186
9.2;6.2 Unconstrained surrogate dual problems for reverse convex infimization;190
9.3;6.3 Constrained surrogate dual problems for reverse convex infimization;199
9.4;6.4 Unperturbational Lagrangian duality for reverse convex infimization;204
9.5;6.5 Duality for infimization over structured primal reverse convex constraint sets;205
10;Optimal Solutions for Reverse Convex Infimization;217
10.1;7.1 Minimum points of functions on reverse convex subsets of locally convex spaces;217
10.2;7.2 Subdifferential characterizations of minimum points of functions on reverse convex sets;223
11;Duality for D.C. Optimization Problems;227
11.1;8.1 Unperturbational duality for unconstrained d. c. infimization;227
11.2;8.2 Minimum points of d. c. functions;235
11.3;8.3 Duality for d. c. infimization with a d. c. inequality constraint;239
11.4;8.4 Duality for d. c. infimization with finitely many d. c. inequality constraints;246
11.5;8.5 Perturbational theory;258
11.6;8.6 Duality for optimization problems involving maximum operators;261
12;Duality for Optimization in the Framework of Abstract Convexity;273
12.1;9.1 Additional preliminaries from abstract convex analysis;273
12.2;9.2 Surrogate duality for abstract quasi- convex supremization, using polarities Ac: 2X --> 2W and Ac: 2 X --> 2W x R;281
12.3;9.3 Constrained surrogate duality for abstract quasi-convex supremization, using families of subsets of X;284
12.4;9.4 Surrogate duality for abstract reverse convex infimization, using polarities AG : 2X -> 2W and AG: 2X -> 2W x R;285
12.5;9.5 Constrained surrogate duality for abstract reverse convex infimization, using families of subsets of X;287
12.6;9.6 Duality for unconstrained abstract d. c. infimization;289
13;Notes and Remarks;292
14;References;341
15;Index;358