Cyclic Relaxed Douglas-Rachford Splitting for Inconsistent Nonconvex Feasibility

TL;DR

This paper investigates a cyclic relaxed Douglas-Rachford algorithm tailored for nonconvex and inconsistent feasibility problems, providing fixed point characterizations and convergence conditions.

Contribution

It introduces a convex relaxation variant of the cyclic Douglas-Rachford algorithm, analyzes its fixed points, and establishes local convergence guarantees in challenging settings.

Findings

Characterized fixed points of the cyclic relaxed Douglas-Rachford algorithm.

Linked fixed points' shadows to those of cyclic projections.

Provided conditions for local convergence in nonconvex, inconsistent cases.

Abstract

We study the cyclic relaxed Douglas-Rachford algorithm for possibly nonconvex, and inconsistent feasibility problems. This algorithm can be viewed as a convex relaxation between the cyclic Douglas-Rachford algorithm first introduced by Borwein and Tam [2014] and the classical cyclic projections algorithm. We characterize the fixed points of the cyclic relaxed Douglas-Rachford algorithm and show the relation of the {\em shadows} of these fixed points to the fixed points of the cyclic projections algorithm. Finally, we provide conditions that guarantee local quantitative convergence estimates in the nonconvex, inconsistent setting.