On the Douglas-Rachford algorithm for solving possibly inconsistent optimization problems

TL;DR

This paper extends the understanding of the Douglas-Rachford algorithm by proving convergence properties of the shadow sequence in cases with disjoint domains, revealing its behavior in more general settings.

Contribution

It establishes the first weak and value convergence results for the shadow sequence with disjoint domains, and analyzes the geometry of the minimal displacement vector.

Findings

Proves weak convergence of the shadow sequence in general settings.

Shows the shadow sequence's limit solves a perturbed normal problem.

Provides new insights into the geometry of the minimal displacement vector.

Abstract

More than 40 years ago, Lions and Mercier introduced in a seminal paper the Douglas-Rachford algorithm. Today, this method is well recognized as a classical and highly successful splitting method to find minimizers of the sum of two (not necessarily smooth) convex functions. While the underlying theory has matured, one case remains a mystery: the behaviour of the shadow sequence when the given functions have disjoint domains. Building on previous work, we establish for the first time weak and value convergence of the shadow sequence generated by the Douglas-Rachford algorithm in a setting of unprecedented generality. The weak limit point is shown to solve the associated normal problem which is a minimal perturbation of the original optimization problem. We also present new results on the geometry of the minimal displacement vector.