Global convergence of a non-convex Douglas-Rachford iteration

TL;DR

This paper proves a global convergence region for a non-convex Douglas-Rachford iteration, improving upon prior local convergence results by explicitly defining where the method reliably finds intersection points.

Contribution

It establishes the first explicit global convergence region for a non-convex Douglas-Rachford iteration involving a line and a circle.

Findings

Proves global convergence for the iteration.

Defines an explicit convergence region.

Extends previous local convergence results.

Abstract

We establish a region of convergence for the proto-typical non-convex Douglas-Rachford iteration which finds a point on the intersection of a line and a circle. Previous work on the non-convex iteration [2] was only able to establish local convergence, and was ineffective in that no explicit region of convergence could be given.