  Douglas-Rachford splitting and its equivalent dual formulation ADMM are
widely used iterative methods in composite optimization problems arising in
control and machine learning applications. The performance of these algorithms
depends on the choice of step size parameters, for which the optimal values are
known in some specific cases, and otherwise are set heuristically. We provide a
new unified method of convergence analysis and parameter selection by
interpreting the algorithm as a linear dynamical system with nonlinear
feedback. This approach allows us to derive a dimensionally independent matrix
inequality whose feasibility is sufficient for the algorithm to converge at a
specified rate. By analyzing this inequality, we are able to give performance
guarantees and parameter settings of the algorithm under a variety of
assumptions regarding the convexity and smoothness of the objective function.
In particular, our framework enables us to obtain a new and simple proof of the
O(1/k) convergence rate of the algorithm when the objective function is not
strongly convex.
