An Inertial Semi-forward-reflected-backward Splitting and Its Application

Inertial methods play a vital role in accelerating the convergence speed of optimization algorithms. This work is concerned with an inertial semi-forward-reflected-backward splitting algorithm of approaching the solution of sum of a maximally monotone operator, a cocoercive operator and a monotone-Lipschitz continuous operator. The theoretical convergence properties of the proposed iterative algorithm are also presented under mild conditions. More importantly, we use an adaptive stepsize rule in our algorithm to avoid calculating Lipschitz constant, which is generally unknown or difficult to estimate in practical applications. In addition, a large class of composite monotone inclusion problem involving mixtures of linearly composed and parallel-sum type monotone operators is solved by combining the primal-dual approach and our proposed algorithm. As a direct application, the obtained inertial algorithm is exploited to composite convex optimization problem and some numerical experiments on image deblurring problem are also investigated to demonstrate the efficiency of the proposed algorithm.