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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. [ABSTRACT FROM AUTHOR]

In this paper, we propose variants of Forward-Backward splitting method for finding a zero of the sum of two operators. A classical modification of Forward-Backward method was proposed by Tseng, which is known to converge when the forward and the backward operators are monotone and with Lipschitz continuity of the forward operator. The conceptual algorithm proposed here improves Tseng’s method in some instances. The first and main part of our approach, contains an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities. During the iteration process, the search performs only one calculation of the forward-backward operator in each tentative of the step. This achieves a considerable computational saving when the forward-backward operator is computationally expensive. The second part of the scheme consists in special projection steps. The convergence analysis of the proposed scheme is given assuming monotonicity on both operators, without Lipschitz continuity assumption on the forward operator. [ABSTRACT FROM AUTHOR]