Duality for Nonconvex Approximation and Optimization (eBook)

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