Relaxed Forward–Backward Splitting Methods for Solving Variational Inclusions and Applications

In this paper, we revisit the modified forward–backward splitting method (MFBSM) for solving a variational inclusion problem of the sum of two operators in Hilbert spaces. First, we introduce a relaxed version of the method (MFBSM) where it can be implemented more easily without the prior knowledge of the Lipschitz constant of component operators. The algorithm uses variable step-sizes which are updated at each iteration by a simple computation. Second, we establish the convergence and the linear rate of convergence of the proposed algorithm. Third, we propose and analyze the convergence of another relaxed algorithm which is a combination between the first one with the inertial method. Finally, we give several numerical experiments to illustrate the convergence of some new algorithms and also to compare them with others.