On the convergence rate of the Douglas-Rachford splitting algorithm

This work is concerned with the convergence rate analysis of the Douglas-Rachford splitting (DRS) method for finding a zero of the sum of two maximally monotone operators. We obtain an exact rate of convergence for the DRS algorithm and demonstrate its sharpness in the setting of convex feasibility problems. Furthermore, we investigate the linear convergence of the DRS algorithm, providing both necessary and sufficient conditions that characterize this behavior. We further examine the performance of the DRS method when applied to convex composite optimization problems. The paper concludes with several conjectures on the convergence behavior of the DRS algorithm for this class of problems.