: Ivan Singer
: Duality for Nonconvex Approximation and Optimization
: Springer-Verlag
: 9780387283951
: 1
: CHF 89.50
:
: Analysis
: English
: 356
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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.
List of Figures10
Contents7
Preface11
Preliminaries18
1.1 Some preliminaries from convex analysis18
1.2 Some preliminaries from abstract convex analysis44
1.3 Duality for best approximation by elements of convex sets56
1.4 Duality for convex and quasi-convex infimization63
Worst Approximation102
2.1 The deviation of a set from an element103
2.2 Characterizations and existence of farthest points110
Duality for Quasi- convex Supremization118
3.1 Some hyperplane theorems of surrogate duality120
3.2 Unconstrained surrogate dual problems for quasi- convex supremization125
3.3 Constrained surrogate dual problems for quasi- convex supremization138
3.4 Lagrangian duality for convex supremization144
3.5 Duality for quasi-convex supremization over structured primal constraint sets148
Optimal Solutions for Quasi- convex Maximization153
4.1 Maximum points of quasi- convex functions153
4.2 Maximum points of continuous convex functions160
4.3 Some basic subdifferential characterizations of maximum points165
Reverse Convex Best Approximation169
5.1 The distance to the complement of a convex set170
5.2 Characterizations and existence of elements of best approximation in complements of convex sets177
Unperturbational Duality for Reverse Convex Infimization184
6.1 Some hyperplane theorems of surrogate duaUty186
6.2 Unconstrained surrogate dual problems for reverse convex infimization190
6.3 Constrained surrogate dual problems for reverse convex infimization199
6.4 Unperturbational Lagrangian duality for reverse convex infimization204
6.5 Duality for infimization over structured primal reverse convex constraint sets205
Optimal Solutions for Reverse Convex Infimization217
7.1 Minimum points of functions on reverse convex subsets of locally convex spaces217
7.2 Subdifferential characterizations of minimum points of functions on reverse convex sets223
Duality for D.C. Optimization Problems227
8.1 Unperturbational duality for unconstrained d. c. infimization227
8.2 Minimum points of d. c. functions235
8.3 Duality for d. c. infimization with a d. c. inequality constraint239
8.4 Duality for d. c. infimization with finitely many d. c. inequality constraints246
8.5 Perturbational theory258
8.6 Duality for optimization problems involving maximum operators261
Duality for Optimization in the Framework of Abstract Convexity273
9.1 Additional preliminaries from abstract convex analysis273
9.2 Surrogate duality for abstract quasi- convex supremization, using polarities Ac: 2X --273
9.2 Surrogate duality for abstract quasi- convex supremization, using polarities Ac: 2X --273
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9.3 Constrained surrogate duality for abstract quasi-convex supremization, using families of subsets of X284
9.4 Surrogate duality for abstract reverse convex infimization, using polarities AG : 2X -284
9.4 Surrogate duality for abstract reverse convex infimization, using polarities AG : 2X -284
285284
9.5 Constrained surrogate duality for abstract reverse convex infimization, using families of subsets of X287
9.6 Duality for unconstrained abstract d. c. infimization289
Notes and Remarks292
References341
Index358