The Douglas-Rachford algorithm in the affine-convex case

TL;DR

This paper extends the convergence theory of the Douglas-Rachford algorithm to cases where one constraint is an affine subspace, broadening its applicability to convex feasibility problems with inconsistent constraints.

Contribution

It provides new convergence results for the Douglas-Rachford algorithm when applied to problems with an affine subspace constraint, generalizing previous results.

Findings

Convergence established for affine subspace constraints.

Extension of Spingarn's result from halfspaces to general convex sets.

Applicable to inconsistent convex feasibility problems.

Abstract

The Douglas-Rachford algorithm is a simple yet effective method for solving convex feasibility problems. However, if the underlying constraints are inconsistent, then the convergence theory is incomplete. We provide convergence results when one constraint is an affine subspace. As a consequence, we extend a result by Spingarn from halfspaces to general closed convex sets admitting least-squares solutions.