Extragradient-Type Methods with $\mathcal{O} (1/k)$ Last-Iterate Convergence Rates for Co-Hypomonotone Inclusions

TL;DR

This paper introduces two accelerated extragradient methods with $(1/k)$ convergence rates for solving co-hypomonotone inclusions, improving theoretical understanding of convergence in such problems.

Contribution

It develops Nesterov's accelerated variants of extragradient methods with proven $(1/k)$ last-iterate convergence rates for co-hypomonotone inclusions, offering new theoretical insights.

Findings

Achieve $
F(1/k)$ convergence rates for residual norm

Require fewer operator evaluations compared to previous methods

Provide new convergence analysis for related gradient-type methods

Abstract

We develop two "Nesterov's accelerated" variants of the well-known extragradient method to approximate a solution of a co-hypomonotone inclusion constituted by the sum of two operators, where one is Lipschitz continuous and the other is possibly multivalued. The first scheme can be viewed as an accelerated variant of Tseng's forward-backward-forward splitting (FBFS) method, while the second one is a Nesterov's accelerated variant of the "past" FBFS scheme, which requires only one evaluation of the Lipschitz operator and one resolvent of the multivalued mapping. Under appropriate conditions on the parameters, we theoretically prove that both algorithms achieve $\mathcal{O}(1/k)$ last-iterate convergence rates on the residual norm, where $k$ is the iteration counter. Our results can be viewed as alternatives of a recent class of Halpern-type methods for root-finding problems. For comparison, we also provide a new convergence analysis of the two recent extra-anchored gradient-type methods for solving co-hypomonotone inclusions.