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

TL;DR

This paper uses performance estimation to derive exact non-ergodic worst-case convergence rates for ADMM, establishing conditions for linear convergence and providing explicit rate formulas.

Contribution

It introduces new non-ergodic convergence bounds for ADMM using performance estimation and characterizes conditions for linear convergence.

Findings

ADMM has a global linear convergence rate under the PL inequality with strong convexity.

Explicit formulas for the linear convergence rate factor are provided.

The paper demonstrates the exactness of the bounds through examples.

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 also study the linear and R-linear convergence of ADMM. We establish that ADMM enjoys a global linear convergence rate if and only if the dual objective satisfies the Polyak-Lojasiewicz (PL)inequality in the presence of strong convexity. In addition, we give an explicit formula for the linear convergence rate factor. Moreover, we study the R-linear convergence of ADMM under two new scenarios.