The exact worst-case convergence rate of the alternating direction method of multipliers

Abstract

Recently, semidefinite programming performance estimation has been employed as a strong tool for the worst-case performance analysis of first order methods. In this paper, we derive new non-ergodic convergence rates for the alternating direction method of multipliers (ADMM) by using performance estimation. We give some examples which show the exactness of the given bounds. We establish that ADMM enjoys a global linear convergence rate if and only if the dual objective satisfies the Polyak-Lojasiewicz (P L) inequality in the presence of strong convexity. In addition, we give an explicit formula for the linear convergence rate factor.