Ioan-Bot / Grad / Wanka Duality in Vector Optimization

1. Auflage 2009
ISBN: 978-3-642-02886-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 400 Seiten

Reihe: Vector Optimization

This book presents fundamentals and comprehensive results regarding duality for scalar, vector and set-valued optimization problems in a general setting. One chapter is exclusively consecrated to the scalar and vector Wolfe and Mond-Weir duality schemes.

Autoren/Hrsg.

Bot, Radu Ioan

Grad, Sorin-Mihai

Wanka, Gert

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;List of symbols and notations;13
4;1 Introduction;16
5;2 Preliminaries on convex analysis and vectoroptimization;23
5.1;2.1 Convex sets;23
5.1.1;2.1.1 Algebraic properties of convex sets;23
5.1.2;2.1.2 Topological properties of convex sets;28
5.2;2.2 Convex functions;33
5.2.1;2.2.1 Algebraic properties of convex functions;33
5.2.2;2.2.2 Topological properties of convex functions;39
5.3;2.3 Conjugate functions and subdifferentiability;44
5.3.1;2.3.1 Conjugate functions;44
5.3.2;2.3.2 Subdifferentiability;52
5.4;2.4 Minimal and maximal elements of sets;56
5.4.1;2.4.1 Minimality;56
5.4.2;2.4.2 Weak minimality;59
5.4.3;2.4.3 Proper minimality;60
5.5;2.5 Vector optimization problems;71
6;3 Conjugate duality in scalar optimization;76
6.1;3.1 Perturbation theory and dual problems;76
6.1.1;3.1.1 The general scalar optimization problem;76
6.1.2;3.1.2 Optimization problems having the composition with a linearcontinuous mapping in the objective function;79
6.1.3;3.1.3 Optimization problems with geometric and cone constraints;81
6.2;3.2 Regularity conditions and strong duality;86
6.2.1;3.2.1 Regularity conditions for the general scalar optimizationproblem;86
6.2.2;3.2.2 Regularity conditions for problems having the compositionwith a linear continuous mapping in the objective function;89
6.2.3;3.2.3 Regularity conditions for problems with geometric and coneconstraints;93
6.3;3.3 Optimality conditions and saddle points;99
6.3.1;3.3.1 The general scalar optimization problem;99
6.4;3.4 The composed convex optimization problem;113
6.4.1;3.4.1 A first dual problem to (PCC);113
6.4.2;3.4.2 A second dual problem to (PCC);118
6.5;3.5 Stable strong duality and formulae for conjugatefunctions and subdifferentials;122
6.5.1;3.5.1 Stable strong duality for the general scalar optimizationproblem;123
6.5.2;3.5.2 The composed convex optimization problem;124
6.5.3;3.5.3 Problems having the composition with a linear continuousmapping in the objective function;127
6.5.4;3.5.4 Problems with geometric and cone constraints;130
7;4 Conjugate vector duality via scalarization;135
7.1;4.1 Fenchel type vector duality;135
7.1.1;4.1.1 Duality with respect to properly efficient solutions;135
7.1.2;4.1.2 Duality with respect to weakly efficient solutions;142
7.2;4.2 Constrained vector optimization: a geometricapproach;144
7.2.1;4.2.1 Duality with respect to properly efficient solutions;144
7.2.2;4.2.2 Duality with respect to weakly efficient solutions;149
7.3;4.3 Constrained vector optimization: a linearscalarization approach;151
7.3.1;4.3.1 A general approach for constructing a vector dual problemvia linear scalarization;152
7.3.2;4.3.2 Vector dual problems to (PV C) as particular instances ofthe general approach;156
7.3.3;4.3.3 The relations between the dual vector problems to (PV C);160
7.3.4;4.3.4 Duality with respect to weakly efficient solutions;165
7.4;4.4 Vector duality via a general scalarization;171
7.4.1;4.4.1 A general duality scheme with respect to a generalscalarization;172
7.4.2;4.4.2 Linear scalarization;177
7.4.3;4.4.3 Maximum(-linear) scalarization;178
7.4.4;4.4.4 Set scalarization;180
7.4.5;4.4.5 (Semi)Norm scalarization;182
7.5;4.5 Linear vector duality;185
7.5.1;4.5.1 The duals introduced via linear scalarization;185
7.5.2;4.5.2 Linear vector duality with respect to weakly efficientsolutions;188
7.5.3;4.5.3 Nakayama’s geometric dual in the linear case;190
8;5 Conjugate duality for vector optimizationproblems with finite dimensional image spaces;193
8.1;5.1 Another Fenchel type vector dual problem;193
8.1.1;5.1.1 Duality with respect to properly efficient solutions;194
8.1.2;5.1.2 Comparisons to (DV A) and (DV ABK);204
8.1.3;5.1.3 Duality with respect to weakly efficient solutions;206
8.2;5.2 A family of Fenchel-Lagrange type vector duals;210
8.2.1;5.2.1 Duality with respect to properly efficient solutions;211
8.2.2;5.2.2 Duality with respect to weakly efficient solutions;221
8.2.3;5.2.3 Duality for linearly constrained vector optimization problems;224
8.3;5.3 Comparisons between different duals to (PV FC);230
8.4;5.4 Linear vector duality for problems with finitedimensional image spaces;239
8.4.1;5.4.1 Duality with respect to properly efficient solutions;239
8.4.2;5.4.2 Duality with respect to weakly efficient solutions;244
8.5;5.5 Classical linear vector duality in finite dimensionalspaces;247
8.5.1;5.5.1 Duality with respect to efficient solutions;247
8.5.2;5.5.2 Duality with respect to weakly efficient solutions;256
9;6 Wolfe and Mond-Weir duality concepts;260
9.1;6.1 Classical scalar Wolfe and Mond-Weir duality;260
9.1.1;6.1.1 Scalar Wolfe and Mond-Weir duality: nondifferentiable case;260
9.1.2;6.1.2 Scalar Wolfe and Mond-Weir duality: differentiable case;262
9.1.3;6.1.3 Scalar Wolfe and Mond-Weir duality under generalizedconvexity hypotheses;265
9.2;6.2 Classical vector Wolfe and Mond-Weir duality;271
9.2.1;6.2.1 Vector Wolfe and Mond-Weir duality: nondifferentiable case;272
9.2.2;6.2.2 Vector Wolfe and Mond-Weir duality: differentiable case;275
9.2.3;6.2.3 Vector Wolfe and Mond-Weir duality with respect to weaklyefficient solutions;280
9.3;6.3 Other Wolfe and Mond-Weir type duals and specialcases;286
9.3.1;6.3.1 Scalar Wolfe and Mond-Weir duality without regularityconditions;287
9.3.2;6.3.2 Vector Wolfe and Mond-Weir duality without regularityconditions;291
9.3.3;6.3.3 Scalar Wolfe and Mond-Weir symmetric duality;294
9.3.4;6.3.4 Vector Wolfe and Mond-Weir symmetric duality;296
9.4;6.4 Wolfe and Mond-Weir fractional duality;301
9.4.1;6.4.1 Wolfe and Mond-Weir duality in scalar fractionalprogramming;301
9.4.2;6.4.2 Wolfe and Mond-Weir duality in vector fractionalprogramming;305
9.5;6.5 Generalized Wolfe and Mond-Weir duality: aperturbation approach;313
9.5.1;6.5.1 Wolfe type and Mond-Weir type duals for general scalaroptimization problems;313
9.5.2;6.5.2 Wolfe type and Mond-Weir type duals for different scalaroptimization problems;314
9.5.3;6.5.3 Wolfe type and Mond-Weir type duals for general vectoroptimization problems;317
10;7 Duality for set-valued optimization problemsbased on vector conjugacy;321
10.1;7.1 Conjugate duality based on efficient solutions;321
10.1.1;7.1.1 Conjugate maps and the subdifferential of set-valued maps;321
10.1.2;7.1.2 The perturbation approach for conjugate duality;329
10.1.3;7.1.3 A special approach - vector k-conjugacy and duality;340
10.2;7.2 The set-valued optimization problem withconstraints;344
10.2.1;7.2.1 Duality based on general vector conjugacy;345
10.2.2;7.2.2 Duality based on vector k-conjugacy;352
10.2.3;7.2.3 Stability criteria;356
10.3;7.3 The set-valued optimization problem having thecomposition with a linear continuous mapping in theobjective function;362
10.3.1;7.3.1 Fenchel set-valued duality;362
10.3.2;7.3.2 Set-valued gap maps for vector variational inequalities;366
10.4;7.4 Conjugate duality based on weakly efficient solutions;370
10.4.1;7.4.1 Basic notions, conjugate maps and subdifferentiability;370
10.4.2;7.4.2 The perturbation approach;376
10.5;7.5 Some particular instances of (PSVGw);382
10.5.1;7.5.1 The set-valued optimization problem with constraints;382
10.5.2;7.5.2 The set-valued optimization problem having the compositionwith a linear continuous mapping in the objective map;387
10.5.3;7.5.3 Set-valued gap maps for set-valued equilibrium problems;389
11;References;394
12;Index;405