The Cyclic Douglas-Rachford Algorithm with r-sets-Douglas-Rachford Operators

TL;DR

This paper introduces a cyclic generalization of the Douglas-Rachford algorithm using r-sets operators, proving convergence and demonstrating potential advantages over traditional methods in convex feasibility problems.

Contribution

It proposes a novel cyclic r-sets-DR operator framework, extending the classical Douglas-Rachford algorithm with theoretical convergence guarantees.

Findings

Convergence of the cyclic r-sets-DR algorithm is proven.

Numerical experiments show potential advantages of r>2 operators.

The method improves performance in convex feasibility problems.

Abstract

The Douglas-Rachford (DR) algorithm is an iterative procedure that uses sequential reflections onto convex sets and which has become popular for convex feasibility problems. In this paper we propose a structural generalization that allows to use $r$-sets-DR operators in a cyclic fashion. We prove convergence and present numerical illustrations of the potential advantage of such operators with $r>2$ over the classical $2$-sets-DR operators in a cyclic algorithm.