Two algorithms for solving systems of inclusion problems

TL;DR

This paper introduces two new algorithms based on forward-backward splitting and hybrid methods for solving systems of inclusion problems involving sums of maximal monotone operators, with convergence guarantees and numerical validation.

Contribution

The paper presents two novel algorithms that improve computational efficiency and do not require Lipschitz continuity, extending existing methods for systems of inclusion problems.

Findings

Algorithms converge under monotonicity without Lipschitz assumption

One algorithm reduces computational cost by fewer projections

Numerical experiments demonstrate effectiveness

Abstract

The goal of this paper is to present two algorithms for solving systems of inclusion problems, with all component of the systems being a sum of two maximal monotone operators. The algorithms are variants of the forward-backward splitting method and one being a hybrid with the alternating projection method. They consist of approximating the solution sets involved in the problem by separating halfspaces which are a well-studied strategy. The schemes contain two part, the first one is an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities. The second part is the projection step, being this the main difference between the algorithms. While the first algorithm computes the projection onto the intersection of the separating halfspaces, the second choose one component of the system and project onto the separating halfspace of this case. In the iterative process, the forward-backward operator is computed once per inclusion problem, representing a relevant computational saving if compared with similar algorithms in the literature. The convergence analysis of the proposed methods is given assuming monotonicity in all operators, without Lipschitz continuity assumption. We also present some numerical experiments.