A Lyapunov function construction for a non-convex Douglas-Rachford iteration

TL;DR

This paper constructs a global Lyapunov function for a non-convex Douglas-Rachford iteration involving intersecting lines, providing a framework for analyzing convergence in complex, non-convex geometries.

Contribution

It introduces a method to combine local Lyapunov functions into a global one for non-convex problems, enhancing convergence analysis tools.

Findings

Global Lyapunov function constructed for non-convex sets

Method applicable to polygonal approximations of complex geometries

Framework suggests robustness in convergence despite non-convexity

Abstract

While global convergence of the Douglas-Rachford iteration is often observed in applications, proving it is still limited to convex and a handful of other special cases. Lyapunov functions for difference inclusions provide not only global or local convergence certificates, but also imply robust stability, which means that the convergence is still guaranteed in the presence of persistent disturbances. In this work, a global Lyapunov function is constructed by combining known local Lyapunov functions for simpler, local sub-problems via an explicit formula that depends on the problem parameters. Specifically, we consider the scenario where one set consists of the union of two lines and the other set is a line, so that the two sets intersect in two distinct points. Locally, near each intersection point, the problem reduces to the intersection of just two lines, but globally the geometry is non-convex and the Douglas-Rachford operator multi-valued. Our approach is intended to be prototypical for addressing the convergence analysis of the Douglas-Rachford iteration in more complex geometries that can be approximated by polygonal sets through the combination of local, simple Lyapunov functions.