Non-Monotone Variational Inequalities

TL;DR

This paper establishes new sufficient conditions for the existence of solutions to non-monotone Variational Inequalities using inverse mapping theory, and analyzes the convergence of the extra-gradient method under these conditions.

Contribution

It introduces novel sufficient conditions for solution existence and convergence in non-monotone VIs, including a condition for Minty solutions and their implications in game theory.

Findings

Extra-gradient method converges to a solution with a Minty solution.

Sufficient conditions for the existence of solutions based on inverse mapping theory.

Weak coupling conditions imply the existence of pure quasi-Nash equilibria.

Abstract

In this paper, we focus on deriving some sufficient conditions for the existence of solutions to non-monotone Variational Inequalities (VIs) based on inverse mapping theory. We have obtained several widely applicable sufficient conditions for this problem and have introduced a sufficient condition for the existence of a Minty solution. We have shown that the extra-gradient method converges to a solution of VI in the presence of a Minty solution. Additionally, we have shown that, under some extra assumption, the algorithm is efficient and approaches a particular type of Minty solution. Interpreting these results in an equivalent game theory problem, weak coupling conditions will be obtained, stating that if the players' cost functions are sufficiently weakly coupled, the game has a pure quasi-Nash equilibrium. Moreover, under the additional assumption of the existence of Minty solutions, a pure Nash equilibrium exists for the corresponding game.