# Circumcentering the Douglas--Rachford method

## TL;DR

The paper introduces the Circumcentered-Douglas-Rachford method, a geometric modification that improves convergence speed for certain feasibility problems involving affine subspaces, outperforming the original method in both theory and practice.

## Contribution

It proposes a new geometric variant of the Douglas-Rachford method with proven convergence properties and better performance, extending applicability to multiple and non-affine convex sets.

## Key findings

- Converges at least as fast as the original Douglas-Rachford method for affine subspaces.
- Linear convergence rate matches the cosine of the Friedrichs angle.
- Numerical results demonstrate improved efficiency over the original method.

## Abstract

We introduce and study a geometric modification of the Douglas-Rach\-ford method called the Circumcentered-Douglas-Rachford method. This method iterates by taking the intersection of bisectors of reflection steps for solving certain classes of feasibility problems. The convergence analysis is established for best approximation problems involving two (affine) subspaces and both our theoretical and numerical results compare favorably to the original Douglas-Rachford method. Under suitable conditions, it is shown that the linear rate of convergence of the Circumcentered-Douglas-Rachford method is at least the cosine of the Friedrichs angle between the (affine) subspaces, which is known to be the sharp rate for the Douglas-Rachford method. We also present a preliminary discussion on the Circumcentered-Douglas-Rachford method applied to the many set case and to examples featuring non-affine convex sets.