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Many large-scale and distributed optimization problems can be brought into acomposite form in which the objective function is given by the sum of a smoothterm and a nonsmooth regularizer. Such problems can be solved via a proximalgradient method and its variants, thereby generalizing gradient descent to anonsmooth setup. In this paper, we view proximal algorithms as dynamicalsystems and leverage techniques from control theory to study their globalproperties. In particular, for problems with strongly convex objectivefunctions, we utilize the theory of integral quadratic constraints to prove theglobal exponential stability of the equilibrium points of the differentialequations that govern the evolution of proximal gradient and Douglas-Rachfordsplitting flows. In our analysis, we use the fact that these algorithms can beinterpreted as variable-metric gradient methods on the suitable envelopes andexploit structural properties of the nonlinear terms that arise from thegradient of the smooth part of the objective function and the proximal operatorassociated with the nonsmooth regularizer. We also demonstrate that theseenvelopes can be obtained from the augmented Lagrangian associated with theoriginal nonsmooth problem and establish conditions for global exponentialconvergence even in the absence of strong convexity.

Proximal gradient flow and Douglas-Rachford splitting dynamics: global exponential stability via integral quadratic constraints