Proximal extrapolated gradient methods for variational inequalities

TL;DR

This paper introduces simple, efficient first-order methods for monotone variational inequalities that do not require Lipschitz continuity and adaptively use local information, often outperforming traditional approaches.

Contribution

The paper proposes novel linesearch-based methods for variational inequalities that are simple, do not need Lipschitz conditions, and are effective even for optimization problems.

Findings

Established ergodic convergence rates for the methods.

Methods perform well with larger stepsizes, improving efficiency.

Simplified linesearch for affine operators requiring only vector-vector multiplication.

Abstract

The paper concerns with novel first-order methods for monotone variational inequalities. They use a very simple linesearch procedure that takes into account a local information of the operator. Also the methods do not require Lipschitz-continuity of the operator and the linesearch procedure uses only values of the operator. Moreover, when operator is affine our linesearch becomes very simple, namely, it needs only vector-vector multiplication. For all our methods we establish the ergodic convergence rate. Although the proposed methods are very general, sometimes they may show much better performance even for optimization problems. The reason for this is that they often can use larger stepsizes without additional expensive computation.