Duality for nonconvex approximation and optimization

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.

Author Notes

Ivan Singer is a Research Professor at the Simion Stoilow Institute of Mathematics in Bucharest, and a Member of the Romanian Academy

List of Figures	p. xi
Preface	p. xiii
1 Preliminaries	p. 1
1.1 Some preliminaries from convex analysis	p. 1
1.2 Some preliminaries from abstract convex analysis	p. 27
1.3 Duality for best approximation by elements of convex sets	p. 39
1.4 Duality for convex and quasi-convex infimization	p. 46
1.4.1 Unperturbational theory	p. 47
1.4.2 Perturbational theory	p. 71
2 Worst Approximation	p. 85
2.1 The deviation of a set from an element	p. 86
2.2 Characterizations and existence of farthest points	p. 93
3 Duality for Quasi-convex Supremization	p. 101
3.1 Some hyperplane theorems of surrogate duality	p. 103
3.2 Unconstrained surrogate dual problems for quasi-convex supremization	p. 108
3.3 Constrained surrogate dual problems for quasi-convex supremization	p. 121
3.4 Lagrangian duality for convex supremization	p. 127
3.4.1 Unperturbational theory	p. 127
3.4.2 Perturbational theory	p. 129
3.5 Duality for quasi-convex supremization over structured primal constraint sets	p. 131
4 Optimal Solutions for Quasi-convex Maximization	p. 137
4.1 Maximum points of quasi-convex functions	p. 137
4.2 Maximum points of continuous convex functions	p. 144
4.3 Some basic subdifferential characterizations of maximum points	p. 149
5 Reverse Convex Best Approximation	p. 153
5.1 The distance to the complement of a convex set	p. 154
5.2 Characterizations and existence of elements of best approximation in complements of convex sets	p. 161
6 Unperturbational Duality for Reverse Convex Infimization	p. 169
6.1 Some hyperplane theorems of surrogate duality	p. 171
6.2 Unconstrained surrogate dual problems for reverse convex infimization	p. 175
6.3 Constrained surrogate dual problems for reverse convex infimization	p. 184
6.4 Unperturbational Lagrangian duality for reverse convex infimization	p. 189
6.5 Duality for infimization over structured primal reverse convex constraint sets	p. 190
6.5.1 Systems	p. 190
6.5.2 Inequality constraints	p. 198
7 Optimal Solutions for Reverse Convex Infimization	p. 203
7.1 Minimum points of functions on reverse convex subsets of locally convex spaces	p. 203
7.2 Subdifferential characterizations of minimum points of functions on reverse convex sets	p. 209
8 Duality for D.C. Optimization Problems	p. 213
8.1 Unperturbational duality for unconstrained d.c. infimization	p. 213
8.2 Minimum points of d.c. functions	p. 221
8.3 Duality for d.c. infimization with a d.c. inequality constraint	p. 225
8.4 Duality for d.c. infimization with finitely many d.c. inequality constraints	p. 232
8.5 Perturbational theory	p. 244
8.6 Duality for optimization problems involving maximum operators	p. 247
8.6.1 Duality via conjugations of type Lau	p. 248
8.6.2 Duality via Fenchel conjugations	p. 252
9 Duality for Optimization in the Framework of Abstract Convexity	p. 259
9.1 Additional preliminaries from abstract convex analysis	p. 259
9.2 Surrogate duality for abstract quasi-convex supremization, using polarities [Delta subscript G]: 2[superscript X] to and [Delta subscript G]: 2[superscript X] to 2[superscript W x R]	p. 267
9.3 Constrained surrogate duality for abstract quasi-convex supremization, using families of subsets of X	p. 270
9.4 Surrogate duality for abstract reverse convex infimization, using polarities [Delta subscript G]: 2[superscript X] to 2[superscript W] and [Delta subscript G]: 2[superscript X] to 2[superscript W x R]	p. 271
9.5 Constrained surrogate duality for abstract reverse convex infimization, using families of subsets of X	p. 273
9.6 Duality for unconstrained abstract d.c. infimization	p. 275
10 Notes and Remarks	p. 279
References	p. 329
Index	p. 347

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