Global Behavior of the Douglas-Rachford Method for a Nonconvex Feasibility Problem

TL;DR

This paper investigates the global behavior of the Douglas-Rachford algorithm in solving nonconvex feasibility problems, especially when intersecting a half-space with a set satisfying certain order or compactness properties, including finite sets.

Contribution

It provides a theoretical analysis of the global convergence behavior of Douglas-Rachford for nonconvex problems under specific set conditions, extending understanding beyond local convergence.

Findings

Analyzes global behavior for sets with well-quasi-ordering or weaker properties.

Includes the case where the second set is finite, relevant for combinatorial optimization.

Offers insights into the algorithm's success in nonconvex settings.

Abstract

In recent times the Douglas-Rachford algorithm has been observed empirically to solve a variety of nonconvex feasibility problems including those of a combinatorial nature. For many of these problems current theory is not sufficient to explain this observed success and is mainly concerned with questions of local convergence. In this paper we analyze global behavior of the method for finding a point in the intersection of a half-space and a potentially non-convex set which is assumed to satisfy a well-quasi-ordering property or a property weaker than compactness. In particular, the special case in which the second set is finite is covered by our framework and provides a prototypical setting for combinatorial optimization problems.