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We provide a new proof of the linear convergence of the alternating directionmethod of multipliers (ADMM) when one of the objective terms is stronglyconvex. Our proof is based on a framework for analyzing optimization algorithmsintroduced in Lessard et al. (2014), reducing algorithm convergence toverifying the stability of a dynamical system. This approach generalizes anumber of existing results and obviates any assumptions about specific choicesof algorithm parameters. On a numerical example, we demonstrate that minimizingthe derived bound on the convergence rate provides a practical approach toselecting algorithm parameters for particular ADMM instances. We complement ourupper bound by constructing a nearly-matching lower bound on the worst-caserate of convergence.

A General Analysis of the Convergence of ADMM