Outer reflected forward–backward splitting algorithm with inertial extrapolation step

This paper studies an outer reflected forward–backward splitting algorithm with an inertial step to find a zero of the sum of three monotone operators composing the maximal monotone operator, Lipschitz monotone operator, and a cocoercive operator in real Hilbert spaces. One of the interesting features of the proposed method is that both the Lipschitz monotone operator and the cocoercive operator are computed explicitly each with one evaluation per iteration. We obtain weak and strong convergence results under some easy-to-verify assumptions. We also obtain a non-asymptotic O(1/n) convergence rate of our proposed algorithm in a non-ergodic sense. We finally give some numerical illustrations arising from compressed sensing and image processing and show that our proposed method is effective and competitive with other related methods in the literature.