Linear convergence of the generalized Douglas-Rachford algorithm for feasibility problems

TL;DR

This paper proves local linear convergence of the generalized Douglas-Rachford algorithm for feasibility problems involving multiple sets, relaxing previous conditions and achieving global convergence in convex cases.

Contribution

It establishes new local linear convergence results for the generalized Douglas-Rachford algorithm, including nonconvex sets, with improved rates and relaxed assumptions.

Findings

Linear convergence holds locally for nonconvex sets.

Global linear convergence in convex cases.

Relaxed regularity conditions compared to previous work.

Abstract

In this paper, we study the generalized Douglas-Rachford algorithm and its cyclic variants which include many projection-type methods such as the classical Douglas-Rachford algorithm and the alternating projection algorithm. Specifically, we establish several local linear convergence results for the algorithm in solving feasibility problems with finitely many closed possibly nonconvex sets under different assumptions. Our findings not only relax some regularity conditions but also improve linear convergence rates in the literature. In the presence of convexity, the linear convergence is global.