On the centralization of the circumcentered-reflection method

TL;DR

This paper introduces a new centralized circumcentered-reflection method (cCRM) that converges globally and linearly for finding intersections of convex sets without product space reformulation, improving efficiency and applicability.

Contribution

The paper develops the first centralized version of CRM that guarantees convergence without product reformulation, with proven linear and superlinear convergence under specific conditions.

Findings

cCRM converges globally for convex set intersections.

cCRM achieves linear convergence under an error bound condition.

Numerical experiments demonstrate the method's effectiveness.

Abstract

This paper is devoted to deriving the first circumcenter iteration scheme that does not employ a product space reformulation for finding a point in the intersection of two closed convex sets. We introduce a so-called centralized version of the circumcentered-reflection method (CRM). Developed with the aim of accelerating classical projection algorithms, CRM is successful for tracking a common point of a finite number of affine sets. In the case of general convex sets, CRM was shown to possibly diverge if Pierra's product space reformulation is not used. In this work, we prove that there exists an easily reachable region consisting of what we refer to as centralized points, where pure circumcenter steps possess properties yielding convergence. The resulting algorithm is called centralized CRM (cCRM). In addition to having global convergence, cCRM converges linearly under an error bound condition, and superlinearly if the two target sets are so that their intersection have nonempty interior and their boundaries are locally differentiable manifolds. We also run numerical experiments with successful results.