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Recently, semidefinite programming performance estimation has been employedas a strong tool for the worst-case performance analysis of first ordermethods. In this paper, we derive new non-ergodic convergence rates for thealternating direction method of multipliers (ADMM) by using performanceestimation. We give some examples which show the exactness of the given bounds.We also study the linear and R-linear convergence of ADMM. We establish thatADMM enjoys a global linear convergence rate if and only if the dual objectivesatisfies the Polyak-Lojasiewicz (PL)inequality in the presence of strongconvexity. In addition, we give an explicit formula for the linear convergencerate factor. Moreover, we study the R-linear convergence of ADMM under two newscenarios.

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