On the linear convergence of the circumcentered-reflection method

TL;DR

This paper proves that the circumcentered-reflection method, which accelerates the Douglas-Rachford method for affine subspaces, maintains linear convergence and can be integrated into classical methods for broader feasibility problems.

Contribution

It extends the circumcentered-reflection method to finitely many affine subspaces and demonstrates its linear convergence, enhancing classical reflection and projection methods.

Findings

The method converges linearly for multiple affine subspaces.

Circumcenters can be embedded into classical reflection and projection methods.

The approach accelerates the best approximation process.

Abstract

In order to accelerate the Douglas--Rachford method we recently developed the circumcentered--reflection method, which provides the closest iterate to the solution among all points relying on successive reflections, for the best approximation problem related to two affine subspaces. We now prove that this is still the case when considering a family of finitely many affine subspaces. This property yields linear convergence and incites embedding of circumcenters within classical reflection and projection based methods for more general feasibility problems.