Ivan Singer Duality for Nonconvex Approximation and Optimization With 17 Figures
نویسندگان
چکیده
except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. To the memory of my wonderful wife, Crina Contents List of Figures xi Preface xiii 1 Preliminaries 1 1.1 Some preliminaries from convex analysis 1 1.2 Some preliminaries from abstract convex analysis 27 1.3 Duality for best approximation by elements of convex sets 39 1.4 Duality for convex and quasi-convex infimization 46 1.4.1 Unperturbational theory 47 1.4.2 Perturbational theory 71 2 Worst Approximation 85 2.1 The deviation of a set from an element 86 2.2 Characterizations and existence of farthest points 93 3 Duality for Quasi-convex Supremization 101 3.1 Some hyperplane theorems of surrogate duality 103 3.2 Unconstrained surrogate dual problems for quasi-convex supremization 108 3.3 Constrained surrogate dual problems for quasi-convex supremization 121 3.4 Lagrangian duality for convex supremization 127 3.4.1 Unperturbational theory 127 3.4.2 Perturbational theory 129 viii Contents 3.5 Duality for quasi-convex supremization over structured primal constraint sets 131 4 Optimal Solutions for Quasi-convex Maximization 137 4.1 Maximum points of quasi-convex functions 137 4.2 Maximum points of continuous convex functions 144 4.3 Some basic subdifferential characterizations of maximum points 149 5 Reverse Convex Best Approximation 153 5.1 The distance to the complement of a convex set 154 5.2 Characterizations and existence of elements of best approximation in complements of convex sets 161 6 Unperturbational Duality for Reverse Convex Infimization 169 6.1 Some hyperplane theorems of surrogate duality 171 6.2 Unconstrained surrogate dual problems for reverse convex infimization 175 6.3 Constrained surrogate dual problems for reverse convex infimization 184 6.4 Unperturbational Lagrangian duality for reverse convex infimization 189 6.5 Duality for infimization over structured primal reverse convex .constraint sets 190 6.5.1 Systems 190 6.5.2 Inequality constraints 198 7 Optimal Solutions for Reverse Convex Infimization 203 7.1 Minimum points of functions on reverse convex subsets of locally convex spaces 203 7.2 Subdifferential characterizations of minimum points of functions on reverse convex sets 209 8 Duality for D.C. Optimization Problems 213 8.1 Unperturbational duality for unconstrained d.c. infimization …
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تاریخ انتشار 2013