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arXiv · 2310.17077

Douglas--Rachford is the best projection method

Minh N. Dao·Mareike Dressler·Hongzhi Liao·Vera Roshchina
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Abstract

We prove that the Douglas--Rachford method applied to two closed convex cones in the Euclidean plane converges in finitely many steps if and only if the set of fixed points of the Douglas--Rachford operator is nontrivial. We analyze this special case using circle dynamics. We also construct explicit examples for a broad family of projection methods for which the set of fixed points of the relevant projection method operator is nontrivial, but the convergence is not finite. This three-parametric family is well known in the projection method literature and includes both the Douglas--Rachford method and the classic method of alternating projections. Even though our setting is fairly elementary, this work contributes in a new way to the body of theoretical research justifying the superior performance of the Douglas--Rachford method compared to other techniques. Moreover, our result leads to a neat sufficient condition for finite convergence of the Douglas--Rachford method in the locally polyhedral case on the plane, unifying and expanding several special cases available in the literature.

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Minh N. Dao, Mareike Dressler, Hongzhi Liao, Vera Roshchina. 2023-10-26. Douglas--Rachford is the best projection method. https://arxiv.org/abs/2310.17077

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