An alternated inertial algorithm with weak and linear convergence for solving monotone inclusions

Inertial-based methods have the drawback of not preserving the Fejer monotonicity of iterative sequences, which may result in slower convergence compared to their corresponding non-inertial versions. To overcome this issue, Mu and Peng [Stat. Optim. Inf. Com-put. 3 (2015), 241-248; MR3393305] suggested an alternating inertial method that can recover the Fejer monotonicity of even sub-sequences. In this paper, we propose a modified version of the forward-backward algorithm with alternating inertial and relaxation effects to solve an inclusion problem in real Hilbert spaces. The weak and linear convergence of the presented algorithm is established under suitable and mild conditions on the involved operators and parameters. Furthermore, the Fejer monotonicity of even subsequences generated by the proposed algorithm with respect to the solution set is recovered. Finally, our tests on image restoration problems demonstrate the superiority of the proposed algorithm over some related results.