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Variational inequalities proved to be a very useful and powerful tool for in vestigation and solution of many equilibrium type problems in Economics, Engineering, Operations Research and Mathematical Physics. In fact, varia tional inequalities for example provide a unifying framework for the study of such diverse problems as boundary value problems, price equilibrium prob lems and traffic network equilibrium problems. Besides, they are closely re lated with many general problems of Nonlinear Analysis, such as fixed point, optimization and complementarity problems. As a result, the theory and so lution methods for variational inequalities have been studied extensively, and considerable advances have been made in these areas. This book is devoted to a new general approach to constructing solution methods for variational inequalities, which was called the combined relax ation (CR) approach. This approach is based on combining, modifying and generalizing ideas contained in various relaxation methods. In fact, each com bined relaxation method has a two-level structure, i.e., a descent direction and a stepsize at each iteration are computed by finite relaxation procedures.
List of contents
1. Variational Inequalities with Continuous Mappings.- 1.1 Problem Formulation and Basic Facts.- 1.2 Main Idea of CR Methods.- 1.3 Implementable CR Methods.- 1.4 Modified Rules for Computing Iteration Parameters.- 1.5 CR Method Based on a Frank-Wolfe Type Auxiliary Procedure.- 1.6 CR Method for Variational Inequalities with Nonlinear Constraints.- 2. Variational Inequalities with Multivalued Mappings.- 2.1 Problem Formulation and Basic Facts.- 2.2 CR Method for the Mixed Variational Inequality Problem.- 2.3 CR Method for the Generalized Variational Inequality Problem.- 2.4 CR Method for Multivalued Inclusions.- 2.5 Decomposable CR Method.- 3. Applications and Numerical Experiments.- 3.1 Iterative Methods for Non Strictly Monotone Variational Inequalities.- 3.2 Economic Equilibrium Problems.- 3.3 Numerical Experiments with Test Problems.- 4 Auxiliary Results.- 4.1 Feasible Quasi-Nonexpansive Mappings.- 4.2 Error Bounds for Linearly Constrained Problems.- 4.3 A Relaxation Subgradient Method Without Linesearch.- Bibliographical Notes.- References.
Summary
Variational inequalities proved to be a very useful and powerful tool for in vestigation and solution of many equilibrium type problems in Economics, Engineering, Operations Research and Mathematical Physics. In fact, varia tional inequalities for example provide a unifying framework for the study of such diverse problems as boundary value problems, price equilibrium prob lems and traffic network equilibrium problems. Besides, they are closely re lated with many general problems of Nonlinear Analysis, such as fixed point, optimization and complementarity problems. As a result, the theory and so lution methods for variational inequalities have been studied extensively, and considerable advances have been made in these areas. This book is devoted to a new general approach to constructing solution methods for variational inequalities, which was called the combined relax ation (CR) approach. This approach is based on combining, modifying and generalizing ideas contained in various relaxation methods. In fact, each com bined relaxation method has a two-level structure, i.e., a descent direction and a stepsize at each iteration are computed by finite relaxation procedures.
