Metric Selection in Douglas-Rachford Splitting and ADMM

Recently, several convergence rate results for Douglas-Rachford splitting and the alternating direction method of multipliers (ADMM) have been presented in the literature. In this paper, we show linear convergence of Douglas-Rachford splitting and ADMM under certain assumptions. We also show that the provided bounds on the linear convergence rates generalize and/or improve on similar bounds in the literature. Further, we show how to select the algorithm parameter to optimize the provided linear convergence rate bound. For smooth and strongly convex finite dimensional problems, we show how the linear convergence rate bounds depend on the metric that is used in the algorithm, and we show how to select this metric to optimize the bound. Since most real-world problems are not both smooth and strongly convex, we also propose heuristic metric and parameter selection methods to improve the performance of a much wider class of problem that not satisfy both these assumptions. These heuristic methods can be applied to problems arising, e.g., in compressed sensing, statistical estimation, model predictive control, and medical imaging. The efficiency of the proposed heuristics is confirmed in a numerical example on a model predictive control problem, where improvements of more than one order of magnitude are observed.