A note on approximate accelerated forward-backward methods with absolute and relative errors, and possibly strongly convex objectives

TL;DR

This paper introduces simplified accelerated forward-backward and hybrid extragradient methods that incorporate approximate computations, handle errors, and leverage strong convexity, with demonstrated numerical performance.

Contribution

It presents a streamlined version of accelerated proximal methods that accommodate errors and strong convexity, extending existing algorithms with practical modifications.

Findings

Methods support both absolute and relative errors.

Numerical experiments illustrate the methods' effectiveness.

Exploitation of strong convexity improves convergence behavior.

Abstract

In this short note, we provide a simple version of an accelerated forward-backward method (a.k.a. Nesterov's accelerated proximal gradient method) possibly relying on approximate proximal operators and allowing to exploit strong convexity of the objective function. The method supports both relative and absolute errors, and its behavior is illustrated on a set of standard numerical experiments. Using the same developments, we further provide a version of the accelerated proximal hybrid extragradient method of Monteiro and Svaiter (2013) possibly exploiting strong convexity of the objective function.