An algorithm for solving the variational inequality problem over the fixed point set of a quasi-nonexpansive operator in Euclidean space

TL;DR

This paper introduces an algorithm for solving variational inequality problems over fixed point sets of quasi-nonexpansive operators, with proven convergence and an application to hierarchical optimization.

Contribution

It proposes a novel projection-based algorithm for VIPs over fixed point sets of quasi-nonexpansive operators, with convergence guarantees.

Findings

Sequences generated by the algorithm converge to the unique VIP solution.

The method is applicable to hierarchical optimization problems.

The algorithm extends existing approaches to a broader class of operators.

Abstract

This paper is concerned with the variational inequality problem (VIP) over the fixed point set of a quasi-nonexpansive operator. We propose, in particular, an algorithm which entails, at each step, projecting onto a suitably chosen half-space, and prove that the sequences it generates converge to the unique solution of the VIP. We also present an application of our result to a hierarchical optimization problem.