Two algorithms for solving systems of inclusion problems

The goal of this paper is to present two algorithms for solving systems of inclusion problems, with all components 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 half-spaces which is a well-studied strategy. The schemes contain two parts, 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, this being the main difference between the algorithms. While the first algorithm computes the projection onto the intersection of the separating half-spaces, the second chooses one component of the system and projects onto the separating half-space 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 of all operators, without Lipschitz continuity assumption. We also present some numerical experiments.