The Douglas--Rachford algorithm for a hyperplane and a doubleton

TL;DR

This paper investigates the behavior of the Douglas--Rachford algorithm when applied to a simple nonconvex inconsistent problem involving a hyperplane and a doubleton, revealing a dependence on rationality of the ratio of distances.

Contribution

It characterizes cycling behavior in this specific nonconvex case and shows how it depends on the rationality of the distance ratio, providing explicit formulas and examples.

Findings

Cycling behavior depends on whether the ratio of distances is rational.

Closed-form expressions for the algorithm's dynamics are derived.

The dynamical behavior exhibits rich and varied patterns.

Abstract

The Douglas--Rachford algorithm is a popular algorithm for solving both convex and nonconvex feasibility problems. While its behaviour is settled in the convex inconsistent case, the general nonconvex inconsistent case is far from being fully understood. In this paper, we focus on the most simple nonconvex inconsistent case: when one set is a hyperplane and the other a doubleton (i.e., a two-point set). We present a characterization of cycling in this case which --- somewhat surprisingly --- depends on whether the ratio of the distance of the points to the hyperplane is rational or not. Furthermore, we provide closed-form expressions as well as several concrete examples which illustrate the dynamical richness of this algorithm.