A text using a new general approach to construction solution methods for variational inequalities, based on combining modifying, and generalizing ideas contained in various relaxation methods. The new approach, termed combined relaxation, is explained and demonstrated. Softcover.
Author(s): Igor Konnov
Series: lecture notes in engeneering economics and Mathematical systems 0495
Publisher: Springer
Year: 2001
Language: English
Pages: 195
Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Preface......Page 8
Table of Contents......Page 10
Notation and Convention......Page 14
1.1.1 Existence and Uniqueness Results......Page 16
1.1.2 Variational Inequalities and Related Problems......Page 21
1.2.1 Newton-like Iterations......Page 28
1.2.2 Basic Properties of CR Methods......Page 32
1.3.1 A CR Method and its Properties......Page 37
1.3.2 Convergence and Rates of Convergence......Page 39
1.3.3 A Modified Line Search......Page 45
1.3.4 Acceleration Techniques for CR Methods......Page 48
1.4.1 A Modified Rule for Computing the Stepsize......Page 51
1.4.2 A Modified Rule for Computing the Descent Direction......Page 55
1.5.1 Description of the Method......Page 60
1.5.2 Convergence......Page 61
1.6.1 A Modified Basic Scheme for CR Methods......Page 63
1.6.2 Description of the Method......Page 64
1.6.3 Convergence......Page 66
2.1.1 Existence and Uniqueness Results......Page 70
2.1.2 Generalized Variational Inequalities and Related Problems......Page 73
2.1.3 Equilibrium and Mixed Variational Inequality Problems......Page 78
2.2.1 Description of the Method......Page 85
2.2.2 Convergence......Page 87
2.2.3 Rate of Convergence......Page 90
2.3.1 Description of the Method......Page 93
2.3.2 Properties of Auxiliary Mappings......Page 95
2.3.3 Convergence......Page 96
2.3.4 Complexity Estimates......Page 98
2.3.5 Modifications......Page 105
2.4.1 An Inexact CR Method for Multivalued Inclusions......Page 108
2.4.2 Convergence......Page 110
2.5 Decomposable CR Method......Page 114
2.5.1 Properties of Auxiliary Mappings......Page 115
2.5.2 Description of the Method......Page 117
2.5.3 Convergence......Page 119
3.1.1 The Proximal Point Method......Page 122
3.1.2 Regularization and Averaging Methods......Page 125
3.1.3 The Ellipsoid Method......Page 127
3.1.4 The Extragradient Method......Page 129
3.2.1 Economic Equilibrium Models as Complementarity Problems......Page 131
3.2.2 Economic Equilibrium Models with Monotone Mappings......Page 132
3.2.3 The Linear Exchange Model......Page 134
3.2.4 The General Equilibrium Model......Page 139
3.2.5 The Oligopolistic Equilibrium Model......Page 141
3.3.1 Systems of Linear Equations......Page 146
3.3.2 Linear Complementarity Problems......Page 147
3.3.3 Linear Variational Inequalities......Page 149
3.3.4 Nonlinear Complementarity Problems......Page 151
3.3.5 Nonlinear Variational Inequalities......Page 153
4.1.1 Exterior Point Algorithms......Page 156
4.1.2 Feasible Point Algorithms......Page 158
4.2.1 Preliminaries......Page 163
4.2.2 Error Bounds......Page 165
4.3.1 Description of the Method......Page 167
4.3.2 Convergence......Page 168
4.3.3 Rate of Convergence......Page 172
Bibliographical Notes......Page 174
References......Page 180
Index......Page 192
