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

We study the generalized forward-reflected-backward (GFRB) method, an extension of the forward-reflected-backward (FRB) scheme due to Malitsky and Tam, for solving monotone inclusion problems in real Hilbert spaces. We first analyze GFRB equipped with a non-decreasing step-size rule that does not require prior knowledge of the Lipschitz constant of the operator involved. We then present two illustrative examples: in the first, we show that the convergence rate of GFRB is bounded from below by that of FRB, and in the second, we obtain an improved convergence rate for GFRB via an appropriate choice of initial parameters. In the sequel, we propose an extended primal-dual twice-reflected (PDTR) algorithm and show that it can be recovered from GFRB under suitable metric selections. Finally, we validate the proposed approach on several state-of-the-art problems and demonstrate better numerical performance compared to the existing ones.