The Douglas-Rachford algorithm for two (not necessarily intersecting) affine subspaces

TL;DR

This paper analyzes the Douglas-Rachford algorithm applied to two affine subspaces that may not intersect, establishing strong convergence of the shadow sequence to the nearest generalized solution, thus extending previous work to inconsistent cases.

Contribution

It provides a detailed convergence analysis of the Douglas-Rachford algorithm for non-intersecting affine subspaces, including new results on the shadow sequence in inconsistent scenarios.

Findings

Shadow sequence converges strongly to the nearest generalized solution.

Extends convergence results from consistent to inconsistent affine subspace cases.

Includes illustrative examples demonstrating the theoretical results.

Abstract

The Douglas--Rachford algorithm is a classical and very successful splitting method for finding the zeros of the sums of monotone operators. When the underlying operators are normal cone operators, the algorithm solves a convex feasibility problem. In this paper, we provide a detailed study of the Douglas--Rachford iterates and the corresponding {shadow sequence} when the sets are affine subspaces that do not necessarily intersect. We prove strong convergence of the shadows to the nearest generalized solution. Our results extend recent work from the consistent to the inconsistent case. Various examples are provided to illustrates the results.