Recently, a worst-case convergence rate was established for the Douglas-Rachford alternating direction method of multipliers (ADMM) in an ergodic sense. The relaxed proximal point algorithm (PPA) is a generalization of the original PPA which includes the Douglas-Rachford ADMM as a special case. In this paper, we provide a simple proof for the same convergence rate of the relaxed PPA in both ergodic and nonergodic senses.
"Convergence Analysis of the Relaxed Proximal Point Algorithm." Abstr. Appl. Anal. 2013 (SI54) 1 - 6, 2013. https://doi.org/10.1155/2013/912846