Forward-Reflected-Backward and Shadow-Douglas--Rachford with partial inverse for Solving Monotone Inclusions

Abstract: In this article, we study two methods for solving monotone inclusions in real Hilbert spaces involving the sum of a maximally monotone operator, a monotone-Lipschitzian operator, a cocoercive operator, and a normal cone to a vector subspace. Our algorithms split and exploits the intrinsic properties of each operator involved in the inclusion. We derive our methods by combining partial inverse techniques with the forward-reflected-backward algorithm and with the shadow-Douglas--Rachford algorithm, respectively. Our methods inherit the advantages of those methods, requiring only one activation of the Lipschitzian operator, one activation of the cocoercive operator, two projections onto the closed vector subspace, and one calculation of the resolvent of the maximally monotone operator. Additionally, to allow larger step-sizes in one of the proposed methods, we revisit FSDR by extending its convergence for larger step-sizes. Furthermore, we provide methods for solving monotone inclusions involving a sum of maximally monotone operators and for solving a system of primal-dual inclusions involving a mixture of sums, linear compositions, parallel sums, Lipschitzian operators, cocoercive operators, and normal cones. We apply our methods to constrained composite convex optimization problems as a specific example. Finally, in order to compare our methods with existing methods in the literature, we provide numerical experiments on constrained total variation least-squares optimization problems and computed tomography inverse problems. We obtain promising numerical results.