Linear Convergence of the Douglas-Rachford Method for Two Closed Sets

TL;DR

This paper proves that the Douglas-Rachford method converges linearly for two closed sets, with global linear convergence in convex cases, under certain regularity conditions.

Contribution

It establishes local R-linear convergence for nonconvex sets and global linear convergence in convex settings, extending recent results.

Findings

Convergence is locally R-linear for nonconvex sets.

Global linear convergence is proven for convex sets.

The results recover and extend recent findings.

Abstract

In this paper, we investigate the Douglas-Rachford method for two closed (possibly nonconvex) sets in Euclidean spaces. We show that under certain regularity conditions, the Douglas-Rachford method converges locally with R-linear rate. In convex settings, we prove that the linear convergence is global. Our study recovers recent results on the same topic.