Chen / Huang / Yang Vector Optimization

Set-valued and Variational Analysis
2005
ISBN: 978-3-540-28445-1
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 541, 308 Seiten

Reihe: Lecture Notes in Economics and Mathematical Systems

Vector optimization model has found many important applications in decision making problems such as those in economics theory, management science, and engineering design (since the introduction of the Pareto optimal solu­ tion in 1896). Typical examples of vector optimization model include maxi­ mization/minimization of the objective pairs (time, cost), (benefit, cost), and (mean, variance) etc. Many practical equilibrium problems can be formulated as variational in­ equality problems, rather than optimization problems, unless further assump­ tions are imposed. The vector variational inequality was introduced by Gi- nessi (1980). Extensive research on its relations with vector optimization, the existence of a solution and duality theory has been pursued. The fundamental idea of the Ekeland's variational principle is to assign an optimization problem a slightly perturbed one having a unique solution which is at the same time an approximate solution of the original problem. This principle has been an important tool for nonlinear analysis and optimization theory. Along with the development of vector optimization and set-valued optimization, the vector variational principle introduced by Nemeth (1980) has been an interesting topic in the last decade. Fan Ky's minimax theorems and minimax inequalities for real-valued func­ tions have played a key role in optimization theory, game theory and math­ ematical economics. An extension was proposed to vector payoffs was intro­ duced by Blackwell (1955).

Autoren/Hrsg.

Chen, Guang-Ya

Huang, Xuexiang

Yang, Xiaogi

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;Introduction and Mathematical Preliminaries;10
3.1;1.1 Convex Cones and Minimal Points;10
3.2;1.2 Elements of Set- Valued Analysis;17
3.3;1.3 Nonlinear Scalarization Functions;21
3.4;1.4 Convex and Generalized Convex Functions;32
3.5;1.5 Notations;41
4;Vector Optimization Problems;45
4.1;2.1 Vector Optimization ( VO);45
4.2;2.2 VO with a Variable Domination Structure;55
4.3;2.3 Characterizations of Solutions for VO;58
4.4;2.4 Continuity of Solutions for VO;68
4.5;2.5 Set-Valued VO with a Fixed Domination Structure;71
4.6;2.6 Set-Valued VO with a Variable Domination Structure;81
4.7;2.7 Augmented Lagrangian Duality for VO;88
4.8;2.8 Augmented Lagrangian Penalization for VO;96
4.9;2.9 Nonlinear Lagrangian Duality for VO;101
4.10;2.10 Nonlinear Penalization for VO;112
5;Vector Variational Inequalities;119
5.1;3.1 Vector Variational Inequalities ( VVI);119
5.2;3.2 Inverse VVI;134
5.3;3.3 Gap Functions for VVI;142
5.4;3.4 Set- valued VVI;149
5.5;3.5 Stability of Generalized Set- valued Quasi- VVI;154
5.6;3.6 Existence of Solutions for Generalized Pre- VVI;159
5.7;3.7 Existence of Solutions for Equilibrium Problems;166
5.8;3.8 Vector Complementarity Problems (VCP);171
5.9;3.9 VCP with a Variable Domination Structure;181
6;Vector Variational Principles;191
6.1;4.1 Variational Principles for Vector- Valued Functions;192
6.2;4.2 Variational Principles for Set- Valued Functions;209
6.3;4.3 Equivalents of Variational Principles for Vector- Valued Functions;222
6.4;4.4 Equivalents of Variational Principles for Set- Valued Functions;230
6.5;4.5 Extended Well- Posedness in Vector- Valued Optimization;237
6.6;4.6 Extended Well- Posedness in Set- Valued Optimization;249
7;Vector Minimax Inequalities;263
7.1;5.1 Minimax Inequalities for Set- Valued Functions;263
7.2;5.2 Minimax Inequalities for Vector- Valued Functions;273
8;Vector Network Equilibrium Problems;279
8.1;6.1 Weak Vector Equilibrium Problem;279
8.2;6.2 Vector Equilibrium Problem;289
8.3;6.3 Dynamic Vector Equilibrium Problem;293
9;References;299
10;Index;310