The subgradient double projection method for variational inequalities in a Hilbert space

Lian Zheng

doi:10.1186/1687-1812-2013-136

Abstract

We present a modification of the double projection algorithm proposed by Solodov and Svaiter for solving variational inequalities (VI) in a Hilbert space. The main modification is to use the subgradient of a convex function to obtain a hyperplane, and the second projection onto the intersection of the feasible set and a halfspace is replaced by projection onto the intersection of two halfspaces. In addition, we propose a modified version of our algorithm that is to find a solution of VI which is also a fixed point of a given nonexpansive mapping. We establish weak convergence theorems for our algorithms.MSC:90C25, 90C30.

Highlights

Let H be a real Hilbert space, C ⊂ H be a nonempty, closed and convex set, and let f : C → H be a mapping
Inspired by the above works, in this paper we present a modification of Algorithm . in a Hilbert space
We propose a modified version of our algorithm that is to find a solution of variational inequalities (VI) which is a fixed point of a given nonexpansive mapping

Summary

Introduction

Let H be a real Hilbert space, C ⊂ H be a nonempty, closed and convex set, and let f : C → H be a mapping. Let ⊂ H be a closed and convex set. Let H be a real Hilbert space, D be a closed and convex subset of H and S : D → H be a nonexpansive mapping. Every closed convex set can be represented in this way, e.g., take c(x) = dist(x, C), where dist is the distance function; see [ , Section .

Results

Conclusion