Gap Functions and Algorithms for Variational Inequality Problems

Congjun Zhang,Jun Wei,Baoqing Liu

doi:10.1155/2013/965640

Abstract

We solve several kinds of variational inequality problems through gap functions, give algorithms for the corresponding problems, obtain global error bounds, and make the convergence analysis. By generalized gap functions and generalized D-gap functions, we give global bounds for the set-valued mixed variational inequality problems. And through gap function, we equivalently transform the generalized variational inequality problem into a constraint optimization problem, give the steepest descent method, and show the convergence of the method.

Highlights

Variational inequality problem (VIP) provides us with a simple, natural, unified, and general frame to study a wide class of equilibrium problems arising in transportation system analysis [1, 2], regional science [3, 4], elasticity [5], optimization [6], and economics [7]
In this paper, we solve some classes of VIP through gap functions, give algorithms for the corresponding problems, obtain global error bounds, and make the convergence analysis
For general variational inequality problem (GVIP), we equivalently transform it into a constraint optimization problem through gap function, introduce the steepest descent method, and show the convergence of the method

Summary

Introduction

Variational inequality problem (VIP) provides us with a simple, natural, unified, and general frame to study a wide class of equilibrium problems arising in transportation system analysis [1, 2], regional science [3, 4], elasticity [5], optimization [6], and economics [7]. Gap functions, which can constitute an equivalent optimization problem, turn out to be very useful in designing new globally convergent algorithms and in analyzing the rate of convergence of some iterative methods. Various gap functions for VIP have been suggested and proposed by many authors in [8, 10,11,12,13] and the references therein. Error bounds play an important role in the analysis of global or local convergence analysis of algorithms for solving VIP. For the VIP defined in (1), the authors in [10] provided an equivalent optimization problem formulation through regularized gap function Gα : H → R defined by. In this paper, we solve some classes of VIP through gap functions, give algorithms for the corresponding problems, obtain global error bounds, and make the convergence analysis. For GVIP, we equivalently transform it into a constraint optimization problem through gap function, introduce the steepest descent method, and show the convergence of the method

Preliminaries

Gap Functions and Error Bounds for SMVIP

Steepest Descent Method for GVIP