On Slater's condition and finite convergence of the Douglas-Rachford algorithm

TL;DR

This paper establishes new conditions under which the Douglas-Rachford algorithm achieves finite convergence for convex feasibility problems, extending previous work and demonstrating practical effectiveness through numerical experiments.

Contribution

It introduces novel sufficient conditions for finite convergence of the Douglas-Rachford algorithm under Slater's condition, expanding its theoretical understanding.

Findings

Finite convergence guaranteed under Slater's condition in specific cases

Numerical results show competitiveness with other methods

Examples illustrate the theoretical results

Abstract

The Douglas-Rachford algorithm is a classical and very successful method for solving optimization and feasibility problems. In this paper, we provide novel conditions sufficient for finite convergence in the context of convex feasibility problems. Our analysis builds upon, and considerably extends, pioneering work by Spingarn. Specifically, we obtain finite convergence in the presence of Slater's condition in the affine-polyhedral and in a hyperplanar-epigraphical case. Various examples illustrate our results. Numerical experiments demonstrate the competitiveness of the Douglas-Rachford algorithm for solving linear equations with a positivity constraint when compared to the method of alternating projections and the method of reflection-projection.