# Outer Approximation Methods for Solving Variational Inequalities in Hilbert Space

## TL;DR

This paper introduces an outer approximation iterative method for solving variational inequalities in Hilbert spaces, extending Fukushima's approach to a more general setting with strong convergence guarantees.

## Contribution

It extends Fukushima's Euclidean method to Hilbert spaces, establishing strong convergence for variational inequalities governed by strongly monotone operators.

## Key findings

- Method achieves strong convergence in Hilbert space.
- Numerical examples demonstrate effectiveness of the approach.
- Applicable to fixed point sets of cutters in variational inequality problems.

## Abstract

In this paper we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $C$. We assume that the set $C$ can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method the main idea of which is to project at each step onto a particular half-space constructed by using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima's method has so far been considered only in the Euclidean setting with different conditions on $F$. We provide several examples for the case where $C$ is the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results.