A variable metric forward--backward method with extrapolation
This work provides a novel optimization algorithm for convex problems with non-differentiable terms, offering faster convergence and broader applicability than existing forward-backward methods.
The paper develops a scaled inertial forward-backward algorithm with variable metric and extrapolation for convex optimization, achieving an O(1/k^2) convergence rate and handling functions with constrained domains. Numerical experiments on image processing, compressed sensing, and statistical inference problems demonstrate effectiveness compared to state-of-the-art methods.
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.