Global exponential stability of the Douglas-Rachford splitting dynamics

Abstract

Many modern optimization problems admit a composite form in which the objective function is given by the sum of a smooth term and a nonsmooth regularizer. Such problems can be solved via proximal methods and their variants, including the Douglas-Rachford (DR) splitting algorithm. In this paper, we view the DR splitting flow as a dynamical system and leverage techniques from control theory to study its global stability properties. In particular, for problems with strongly convex objective functions, we utilize the theory of integral quadratic constraints to prove global exponential stability of the ordinary differential equation that governs the evolution of the DR splitting flow. In our analysis, we use the fact that this algorithm can be interpreted as a variable-metric gradient method on the DR envelope and exploit structural properties of nonlinear terms that arise from composition of reflected proximal operators.