Two Relaxed Inertial Forward–Reflected–Backward Splitting Algorithms With Momentum Terms

In this paper, to solve the monotone inclusion problem consisting of the sum of two monotone operators in Hilbert spaces, we propose and study two modifications of Malitsky–Tam’s forward–reflection–backward splitting methods with double momentum terms. Meanwhile, we consider a relaxed inertial version to expand the range of allowable step sizes. Under the same assumptions as the Malitsky–Tam’s method (i.e., the set‐valued operator is maximally monotone, and the single‐valued operator is Lipschitz continuous and monotone), we prove the weak convergence and linear convergence of the proposed methods, respectively. Numerical results show that the relaxed inertial version effectively improves the convergence performance compared to the Malitsky–Tam’s splitting algorithm and its inertial version.