A strongly convergent inertial inexact proximal-point algorithm for monotone inclusions with applications to variational inequalities

TL;DR

This paper introduces a new inertial inexact proximal-point algorithm with strong convergence guarantees for monotone inclusions, extending its applicability to variational inequalities and related methods, with promising preliminary numerical results.

Contribution

It develops a strongly convergent inertial proximal-point algorithm with less restrictive assumptions, applicable to variational inequalities and related methods, improving convergence analysis.

Findings

Proved strong convergence under weaker assumptions.

Extended the algorithm to variational inequalities.

Preliminary numerical experiments show good performance.

Abstract

We propose an inertial variant of the strongly convergent inexact proximal-point (PP) method of Solodov and Svaiter (2000) for monotone inclusions. We prove strong convergence of our main algorithm under less restrictive assumptions on the inertial parameters when compared to previous analysis of inertial PP-type algorithms, which makes our method of interest even in finite-dimensional settings. We also performed an iteration-complexity analysis and applied our main algorithm to variational inequalities for monotone operators, obtaining strongly convergent (inertial) variants of Korpolevich's extragradient, forward-backward and Tseng's modified forward-backward methods. Preliminary numerical experiments indicate that our strongly convergent variant of Tseng's modified forward-backward method performs well on certain matrix game problems.