Relaxed and inertial nonlinear Forward-Backward algorithm

Abstract: The Nonlinear Forward-Backward (NFB) algorithm, also known as warped resolvent iterations, is a splitting method for finding zeros of sums of monotone operators. In particular cases, NFB reduces to well-known algorithms such as Forward-Backward, Forward-Backward-Forward, Chambolle--Pock, and Condat-Vu. Therefore, NFB can be used to solve monotone inclusions involving sums of maximally monotone, cocoercive, monotone and Lipschitz operators as well as linear compositions terms. In this article, we study the convergence of NFB with inertial and relaxation steps. Our results recover known convergence guarantees for the aforementioned methods when extended with inertial and relaxation terms. Additionally, we establish the convergence of inertial and relaxed variants of the Forward-Backward-Half-Forward and Forward-Primal--Dual-Half-Forward algorithms, which, to the best of our knowledge, are new contributions. We consider both nondecreasing and decreasing sequences of inertial parameters, the latter being a novel approach in the context of inertial algorithms. To evaluate the performance of these strategies, we present numerical experiments on optimization problems with affine constraints and on image restoration tasks. Our results show that decreasing inertial sequences can accelerate the numerical convergence of the algorithms.