Multi-step inertial forward-backward-half forward algorithm for solving monotone inclusion

In this article, we study a multi-step inertial forward-backward-half forward splitting algorithm to solve the solution of the sum of three operators with two of them being respectively, cocoercive and monotone-Lipschitz continuous. Notably, the convergence of the multi-step inertial forward-backward-half forward algorithm is established under the summability condition formulated on the basis of the iterative sequence in a Hilbert space setting. We also display two applications including problems that are: a composite monotone inclusion problem involving mixtures of linearly composed with parallel-sum type operators, and a composite convex optimization problem. As far as the resolvent of the composite operator is concerned, due to difficulty in calculation, then, the composite monotone inclusion problem is addressed by the proposed multi-step inertial algorithm together with the primal-dual idea. Experiments conducted on an image deblurring problem indicate the superiority and efficiency of the proposed method.