Continuous and Discrete Systems for Quasi Variational Inequalities with Application to Game Theory

TL;DR

This paper introduces a new class of third-order projected dynamical systems for quasi variational inequalities, analyzing their stability and proposing iterative schemes with convergence guarantees, with applications to generalized Nash equilibrium problems.

Contribution

It develops a novel third-order dynamical system framework for quasi variational inequalities and proposes convergent iterative schemes with applications to game theory.

Findings

Stability analysis of the continuous gradient-type method.

Development of implicit and explicit iterative schemes.

Application to generalized Nash equilibrium problems.

Abstract

A new class of projected dynamical systems of third order is investigated for quasi (parametric) variational inequalities in which the convex set in the classical variational inequality also depends upon the solution explicitly or implicitly. We study the stability of a continuous method of a gradient type. Some iterative implicit and explicit schemes are suggested as counterparts of the continuous case by inertial proximal methods. The convergence analysis of these proposed methods is established under sufficient mild conditions. Moreover, some applications dealing with the generalized Nash equilibrium problems are presented.