Vector Optimization (eBook)

Set-valued and Variational Analysis

Guang-ya Chen, Xuexiang Huang, Xiaogi Yang (Autoren)

eBook Download: PDF

2005 | 2005
X, 308 Seiten
Springer Berlin (Verlag)
978-3-540-28445-1 (ISBN)

Infos

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).

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 20.11.2005
• Reihe/Serie: Lecture Notes in Economics and Mathematical Systems
• Reihe/Serie: Lecture Notes in Economics and Mathematical Systems
• Zusatzinfo: X, 308 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Allgemeines / Lexika
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Planung / Organisation
• Themenwelt: Wirtschaft ► Volkswirtschaftslehre
• Schlagworte: Analysis • Operations Research • Optimization • Scalarization Method • Vector Equilibrium Problem • Vector Optimization • Vector Variational Inequality • Vector Variational Principle
• ISBN-10: 3-540-28445-1 / 3540284451
• ISBN-13: 978-3-540-28445-1 / 9783540284451