An Approximation-Based Regularized Extra-Gradient Method for Monotone Variational Inequalities

TL;DR

This paper introduces a versatile regularized extra-gradient method for monotone variational inequalities that unifies various solution approaches and achieves optimal convergence rates, especially for strongly monotone problems.

Contribution

It proposes a general approximation-based regularized extra-gradient framework that encompasses multiple methods and guarantees optimal convergence, including superlinear local convergence for strongly monotone VI.

Findings

Global convergence of ARE is guaranteed.

Superlinear local convergence for strongly monotone VI.

Framework unifies various solution methods.

Abstract

In this paper, we propose a general extra-gradient scheme for solving monotone variational inequalities (VI), referred to here as Approximation-based Regularized Extra-gradient method (ARE). The first step of ARE solves a VI subproblem with an approximation operator satisfying a $p^{th}$-order Lipschitz bound with respect to the original mapping, further coupled with the gradient of a $(p+1)^{th}$-order regularization. The optimal global convergence is guaranteed by including an additional extra-gradient step, while a $p^{th}$-order superlinear local convergence is shown to hold if the VI is strongly monotone. The proposed ARE is inclusive and general, in the sense that a variety of solution methods can be formulated within this framework as different manifestations of approximations, and their iteration complexities would follow through in a unified fashion. The ARE framework relates to the first-order methods, while opening up possibilities to developing higher-order methods specifically for structured problems that guarantee the optimal iteration complexity bounds.