On Generalized Forward-Reflected-Backward Method for Monotone Inclusion Problems

Abstract: In this article, we analyze the generalized forward-reflected-backward (GFRB) method, which is an extension of the forward-reflected-backward (FRB) method due to Malitsky and Tam, for solving monotone inclusion problems in real Hilbert spaces. In particular, we study GFRB with an increasing step-size update that does not require prior knowledge of the Lipschitz constant to run the algorithm. In this sequel, we propose an extended version of the primal-dual twice-reflected algorithm (PDTR), which can be derived from GFRB when some metrics are chosen appropriately. Furthermore, for a particular example, we prove that the rate of convergence of GFRB is tight to FRB, and in another example, we improve the rate of convergence of GFRB compared to existing algorithms by choosing proper initial parameters. Finally, we validate the proposed approach on several state-of-the-art problems and demonstrate better numerical performance compared to the existing ones.