Outer Approximation Methods for Solving Variational Inequalities Defined over the Solution Set of a Split Convex Feasibility Problem

TL;DR

This paper introduces outer approximation methods for solving variational inequalities over the solution set of a split convex feasibility problem, using modified gradient projection techniques and Landweber transform.

Contribution

It proposes three variants of a method that replace metric projections with half-space projections, leveraging the problem's split structure and Landweber transform.

Findings

Developed three new outer approximation algorithms.

Proved convergence of the proposed methods.

Utilized Landweber transform for split structure handling.

Abstract

We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $S$. We assume that $S=C\cap A^{-1}(Q)$ is the nonempty solution set of a (multiple-set) split convex feasibility problem, where $C$ and $Q$ are both closed and convex subsets of two real Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$, respectively, and the operator $A$ acting between them is linear. We consider a modification of the gradient projection method the main idea of which is to replace at each step the metric projection onto $S$ by another metric projection onto a half-space which contains $S$. We propose three variants of a method for constructing the above-mentioned half-spaces by employing the multiple-set and the split structure of the set $S$. For the split part we make use of the Landweber transform.