Diagonal Scaling in Douglas-Rachford Splitting and ADMM

The performance of Douglas-Rachford splitting and the alternating direction method of multipliers (ADMM) (i.e. Douglas-Rachford splitting on the dual problem) are sensitive to conditioning of the problem data. For a restricted class of problems that enjoy a linear rate of convergence, we show in this paper how to precondition the optimization data to optimize a bound on that rate. We also generalize the preconditioning methods to problems that do not satisfy all assumptions needed to guarantee a linear convergence. The efficiency of the proposed preconditioning is confirmed in a numerical example, where improvements of more than one order of magnitude are observed compared to when no preconditioning is used.