Local Convergence Properties of Douglas--Rachford and ADMM

TL;DR

This paper analyzes the local linear convergence of Douglas--Rachford and ADMM algorithms when applied to partly smooth convex functions, showing finite identification of manifolds and convergence rates.

Contribution

It provides new theoretical insights into the local convergence behavior of DR/ADMM for partly smooth functions, including finite manifold identification and explicit convergence rates.

Findings

DR/ADMM identify smooth manifolds in finite time

Local linear convergence occurs after manifold identification

Convergence radius depends on the Friedrichs angle between tangent spaces

Abstract

The Douglas--Rachford (DR) and alternating direction method of multipliers (ADMM) are two proximal splitting algorithms designed to minimize the sum of two proper lower semi-continuous convex functions whose proximity operators are easy to compute. The goal of this work is to understand the local linear convergence behaviour of DR/ADMM when the involved functions are moreover partly smooth. More precisely, when the two functions are partly smooth relative to their respective smooth submanifolds, we show that DR/ADMM (i) identifies these manifolds in finite time; (ii) enters a local linear convergence regime. When both functions are locally polyhedral, we show that the optimal convergence radius is given in terms of the cosine of the Friedrichs angle between the tangent spaces of the identified submanifolds. Under polyhedrality of both functions, we also provide condition sufficient for finite convergence of DR. The obtained results are illustrated by several concrete examples and supported by numerical experiments.