An algorithm for solving a multi-valued variational inequality

Changjie Fang,Chunde Yang,Shenglan Chen

doi:10.1186/1029-242x-2013-218

Abstract

We propose a new extragradient method for solving a multi-valued variational inequality. It is showed that the method converges globally to a solution of the multi-valued variational inequality, provided the multi-valued mapping is continuous with nonempty compact convex values. Preliminary computational experience is also reported. MSC: 47H04; 47H10; 47J20; 47J25

Highlights

We consider the following multi-valued variational inequality, denoted by MVI(F, C): to find x* ∈ C and ξ ∈ F(x*) such that ξ, y – x* ≥, ∀y ∈ C, ( . )where C is a nonempty closed convex set in Rn, F is a multi-valued mapping from C into Rn with nonempty values, and ·, · and · denote the inner product and the norm in Rn, respectively.Extragradient-type algorithms have been extensively studied in the literature; see [ – ]
Where C is a nonempty closed convex set in Rn, F is a multi-valued mapping from C into Rn with nonempty values, and ·, · and · denote the inner product and the norm in Rn, respectively
Various algorithms for solving the multi-valued variational inequality have been extensively studied in the literature [ – ]

Summary

Introduction

We consider the following multi-valued variational inequality, denoted by MVI(F, C): to find x* ∈ C and ξ ∈ F(x*) such that ξ , y – x* ≥ , ∀y ∈ C, ( . )where C is a nonempty closed convex set in Rn, F is a multi-valued mapping from C into Rn with nonempty values, and ·, · and · denote the inner product and the norm in Rn, respectively.Extragradient-type algorithms have been extensively studied in the literature; see [ – ]. Introduction We consider the following multi-valued variational inequality, denoted by MVI(F, C): to find x* ∈ C and ξ ∈ F(x*) such that ξ , y – x* ≥ , ∀y ∈ C, Various algorithms for solving the multi-valued variational inequality have been extensively studied in the literature [ – ].

Results

Conclusion