A projection algorithm for non-monotone variational inequalities

TL;DR

This paper presents a projection algorithm for solving non-monotone variational inequalities without requiring monotonicity, proving convergence under continuity assumptions and demonstrating its effectiveness through numerical experiments.

Contribution

It introduces a new projection-type algorithm for non-monotone variational inequalities with proven convergence, expanding applicability beyond monotone cases.

Findings

Whole sequence convergence to a solution

Algorithm performs well in numerical experiments

Comparison shows competitive performance with recent methods

Abstract

We introduce and study the convergence properties of a projection-type algorithm for solving the variational inequality problem for point-to-set operators. No monotoni\-city assumption is used in our analysis. The operator defining the problem is only assumed to be continuous in the point-to-set sense, i.e., inner- and outer-semicontinuous. Additionally, we assume non-emptiness of the so-called dual solution set. We prove that the whole sequence of iterates converges to a solution of the variational inequality. Moreover, we provide numerical experiments illustrating the behavior of our iterates. Through several examples, we provide a comparison with a recent similar algorithm.