On the Minimal Displacement Vector of the Douglas-Rachford Operator

TL;DR

This paper studies the minimal displacement vector of the Douglas-Rachford operator, providing new properties that help understand divergence and infeasibility in optimization problems.

Contribution

It introduces generalized properties of the minimal displacement vector, extending previous results and enhancing understanding of the operator's behavior when fixed points do not exist.

Findings

Minimal displacement vector converges even when fixed points do not exist.

New properties of the minimal displacement vector are established.

Results help certify infeasibility in optimization problems.

Abstract

The Douglas-Rachford algorithm can be represented as the fixed point iteration of a firmly nonexpansive operator. When the operator has no fixed points, the algorithm's iterates diverge, but the difference between consecutive iterates converges to the so-called minimal displacement vector, which can be used to certify infeasibility of an optimization problem. In this paper, we establish new properties of the minimal displacement vector, which allow us to generalize some existing results.