Q-quadratic convergence of the centralized circumcentered-reflection method under a relative interior condition

TL;DR

This paper proves that the centralized circumcentered-reflection method (cCRM) converges superlinearly or quadratically under certain relative interior and boundary smoothness conditions, extending prior convergence results.

Contribution

It establishes superlinear and quadratic convergence of cCRM in cases where the affine hulls coincide and boundary smoothness conditions are met, including explicit asymptotic constants.

Findings

cCRM converges superlinearly when relative interiors intersect and boundaries are  hypersurfaces.

cCRM converges Q-quadratically when boundaries are   smooth with explicit constants.

The case where affine hulls differ remains an open problem.

Abstract

The centralized circumcentered-reflection method (\cCRM) of~\cite{Behling:2024} converges superlinearly to a solution of $\operatorname{find}\;z\in X\cap Y$ when $\inte(X\cap Y)\neq\emptyset$ and the boundaries of $X$ and $Y$ are $\mathcal{C}^1$ hypersurfaces in $\re^n$. Both conditions fail when $\aff(X)=\aff(Y)\subsetneq\re^n$, as in equality-constrained feasibility and spectral matrix problems. We prove that \cCRM\ converges superlinearly when $\aff(X)=\aff(Y)$, $\operatorname{ri}(X)\cap\operatorname{ri}(Y)\neq\emptyset$, and the relative boundaries are $\mathcal{C}^1$ of appropriate relative dimension; and Q-quadratically when the relative boundaries are $\mathcal{C}^2$, with explicit asymptotic constant expressed in terms of the boundary curvatures at the limit point and the local error-bound constant. The case $\aff(X)\neq\aff(Y)$ is identified as open.