Finite Convergence of Circumcentered-Reflection Method on Closed Polyhedral Cones in Euclidean Spaces

TL;DR

This paper proves that the Circumcentered Reflection Method (CRM) finitely converges to a feasible point in the intersection of convex cones and polyhedral sets in Euclidean spaces, improving understanding of its convergence properties.

Contribution

It establishes finite convergence of CRM for intersections of convex cones and polyhedral sets, introduces a modified Sphere-Centered Reflection Method, and analyzes conditions affecting convergence.

Findings

CRM finitely converges in \\mathbb{R}^2 for convex cones.

CRM and its variant can find feasible points in finitely many steps.

Finite convergence may fail outside specific initial regions.

Abstract

The Circumcentered Reflection Method (CRM) is a recently developed projection method for solving convex feasibility problems. It offers preferable convergence properties compared to classic methods such as the Douglas-Rachford and the alternating projections method. In this study, our first main theorem establishes that CRM can identify a feasible point in the intersection of two closed convex cones in \(\mathbb{R}^2\) from any starting point in the Euclidean plane. We then apply this theorem to intersections of two polyhedral sets in \(\mathbb{R}^2\) and two wedge-like sets in \(\mathbb{R}^n\), proving that CRM converges to a point in the intersection from any initial position finitely. Additionally, we introduce a modified technique based on CRM, called the Sphere-Centered Reflection Method. With the help of this technique, we demonstrate that CRM can locate a feasible point in finitely many iterations in the intersection of two proper polyhedral cones in \(\mathbb{R}^3\) when the initial point lies in a subset of the complement of the intersection's polar cone. Lastly, we provide an example illustrating that finite convergence may fail for the intersection of two proper polyhedral cones in \(\mathbb{R}^3\) if the initial guess is outside the designated set.