Activity Identification and Local Linear Convergence of Douglas–Rachford / ADMM under Partial Smoothness

Abstract

LNCS n°9087Proximal splitting algorithms are becoming popular to solve convex optimization prob-lems in variational image processing. Within this class, Douglas–Rachford (DR) and ADMMare designed to minimize the sum of two proper lower semicontinuous convex functionswhose proximity operators are easy to compute. The goal of this work is to understandthe local convergence behaviour of DR/ADMM when the involved functions are moreoverpartly smooth. More precisely, when one of the functions and the conjugate of the otherare partly smooth relative to their respective manifolds, we show that DR/ADMM identifiesthese manifolds in finite time. Moreover, when these manifolds are affine or linear, we provethat DR is locally linearly convergent with a rate in terms of the cosine of the Friedrichsangle between the two tangent subspaces. This is illustrated by several concrete examplesand supported by numerical experiments.nonnonouirechercheInternationa