Vector Optimization: Set-valued and Variational Analysis (Lecture Notes in Economics and Mathematical Systems)

✍ Scribed by Guang-ya Chen, Xuexiang Huang, Xiaogi Yang

• Year: 2005
• Tongue: English
• Leaves: 315
• Edition: 1
• Category: Library

✦ Synopsis

This book is devoted to vector or multiple criteria approaches in optimization. Topics covered include: vector optimization, vector variational inequalities, vector variational principles, vector minmax inequalities and vector equilibrium problems. In particular, problems with variable ordering relations and set-valued mappings are treated. The nonlinear scalarization method is extensively used throughout the book to deal with various vector-related problems. The results presented are original and should be interesting to researchers and graduates in applied mathematics and operations research. Readers will benefit from new methods and ideas for handling multiple criteria decision problems.

✦ Table of Contents

Contents......Page 8
1.1 Convex Cones and Minimal Points......Page 10
1.2 Elements of Set-Valued Analysis......Page 17
1.3 Nonlinear Scalarization Functions......Page 21
1.4 Convex and Generalized Convex Functions......Page 32
1.5 Notations......Page 41
2.1 Vector Optimization (VO)......Page 45
2.2 VO with a Variable Domination Structure......Page 55
2.3 Characterizations of Solutions for VO......Page 58
2.4 Continuity of Solutions for VO......Page 68
2.5 Set-Valued VO with a Fixed Domination Structure......Page 71
2.6 Set-Valued VO with a Variable Domination Structure......Page 81
2.7 Augmented Lagrangian Duality for VO......Page 88
2.8 Augmented Lagrangian Penalization for VO......Page 96
2.9 Nonlinear Lagrangian Duality for VO......Page 101
2.10 Nonlinear Penalization for VO......Page 112
3.1 Vector Variational Inequalities (VVI)......Page 119
3.2 Inverse VVI......Page 134
3.3 Gap Functions for VVI......Page 142
3.4 Set-valued VVI......Page 149
3.5 Stability of Generalized Set-valued Quasi-VVI......Page 154
3.6 Existence of Solutions for Generalized Pre-VVI......Page 159
3.7 Existence of Solutions for Equilibrium Problems......Page 166
3.8 Vector Complementarity Problems (VCP)......Page 171
3.9 VCP with a Variable Domination Structure......Page 181
4 Vector Variational Principles......Page 191
4.1 Variational Principles for Vector-Valued Functions......Page 192
4.2 Variational Principles for Set-Valued Functions......Page 209
4.3 Equivalents of Variational Principles for Vector-Valued Functions......Page 222
4.4 Equivalents of Variational Principles for Set-Valued Functions......Page 230
4.5 Extended Well-Posedness in Vector-Valued Optimization......Page 237
4.6 Extended Well-Posedness in Set-Valued Optimization......Page 249
5.1 Minimax Inequalities for Set-Valued Functions......Page 263
5.2 Minimax Inequalities for Vector-Valued Functions......Page 273
6.1 Weak Vector Equilibrium Problem......Page 279
6.2 Vector Equilibrium Problem......Page 289
6.3 Dynamic Vector Equilibrium Problem......Page 293
References......Page 299
D......Page 310
N......Page 311
W......Page 312
Z......Page 313