A Lyapunov Function Construction for a Non-convex Douglas–Rachford Iteration

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.2010 Mathematics Subject Classification. 47H10, 47J25, 37N40, 90C26.

Circumcentering Reflection Methods for Nonconvex Feasibility Problems

“…Spheres and subspaces are of interest because they are prototypical of phase retrieval. They were studied for the Douglas-Rachford method in [2,18,20], and the Lyapunov function discovered in [18] has been catalytic in other nonconvex investigations [24,33]. Interestingly, global convergence for CRM for spheres and hyperplanes is implicitly shown in [16, see Remark 1].…”

Computable Centering Methods for Spiraling Algorithms and their Duals, with Motivations from the theory of Lyapunov Functions

“…Other works that use Lyapunov functions for studying DR iteration have constructed them explicitly, in order to guarantee robust KL-stability on an explicit region U by using Theorem 2.10. This was true of the works of Benoist [11], Dao and Tam [19], and Giladi and Rüffer [24]. The goal of this section is the converse of this approach.…”

“…We will use robust KL-stability, together with results of Kellett and Teel [30], to show the existence of Lyapunov functions that describe the behaviour of Douglas-Rachford method in many settings. The following introduction to Lyapunov functions and KLstability is quite standard and closely follows those in the works of Giladi and Rüffer [24] and of Kellett and Teel [30].…”

“…Giladi and Rüffer used this result to guarantee robust KL-stability for their setting, by means of constructing the prerequisite Lyapunov function [24].…”

“…Dao and Tam extended Benoist's approach to function graphs more generally [19]. Most recently, Giladi and Rüffer broadly demonstrated the power of this approach by using local Lyapunov functions to construct a global Lyapunov function in order to show robust KL-stability of DR for solving (3) when one set is a line and the other set is the union of two lines [24]. Altogether, Lyapunov functions have become the definitive approach to describing the basins of attraction for DR in the setting of nonconvex feasibility problems.…”