A variable metric forward--backward method with extrapolation

TL;DR

This paper introduces a scaled inertial forward-backward algorithm with extrapolation for convex optimization, capable of handling functions with restricted domains, achieving accelerated convergence and demonstrating superior performance in image processing, compressed sensing, and statistical inference.

Contribution

It develops a novel variable metric inertial forward-backward method with extrapolation that handles non-entire domain functions and proves accelerated convergence rates.

Findings

Achieves an ${ m O}(1/k^2)$ convergence rate.

Proves convergence of the iterates.

Outperforms existing algorithms in numerical tests.

Abstract

Forward-backward methods are a very useful tool for the minimization of a functional given by the sum of a differentiable term and a nondifferentiable one and their investigation has experienced several efforts from many researchers in the last decade. In this paper we focus on the convex case and, inspired by recent approaches for accelerating first-order iterative schemes, we develop a scaled inertial forward-backward algorithm which is based on a metric changing at each iteration and on a suitable extrapolation step. Unlike standard forward-backward methods with extrapolation, our scheme is able to handle functions whose domain is not the entire space. Both {an ${\mathcal O}(1/k^2)$ convergence rate estimate on the objective function values and the convergence of the sequence of the iterates} are proved. Numerical experiments on several {test problems arising from image processing, compressed sensing and statistical inference} show the {effectiveness} of the proposed method in comparison to well performing {state-of-the-art} algorithms.