A remark on the convergence of the Douglas-Rachford iteration in a non-convex setting

TL;DR

This paper demonstrates that the Douglas-Rachford iteration, when applied to a sphere and a line in high-dimensional space, exhibits robust convergence properties using Lyapunov functions, extending understanding in non-convex optimization.

Contribution

It introduces a Lyapunov function-based approach to prove robust convergence of Douglas-Rachford iteration in a non-convex setting involving a sphere and a line.

Findings

Douglas-Rachford iteration is robustly a4L-stable for a sphere and a line

Convergence is stronger than uniform convergence on compact sets

Lyapunov functions can be used to analyze non-convex convergence

Abstract

Using the construction of a Lyapunov function, it is shown that the Douglas-Rachford iteration with respect to a sphere and a line in $\mathbb R^d$ is robustly $\mathcal{KL}$-stable. This implies a convergence which is stronger than uniform convergence on compact sets.