The Block-wise Circumcentered-Reflection Method

Abstract

The elementary Euclidean concept of circumcenter has recently been employed to improve two aspects of the classical Douglas--Rachford method for projecting onto the intersection of affine subspaces. The so-called circumcentered-reflection method is able to both accelerate the average reflection scheme by the Douglas--Rachford method and cope with the intersection of more than two affine subspaces. We now introduce the technique of circumcentering in blocks, which, more than just an option over the basic algorithm of circumcenters, turns out to be an elegant manner of generalizing the method of alternating projections. Linear convergence for this novel block-wise circumcenter framework is derived and illustrated numerically. Furthermore, we prove that the original circumcentered-reflection method essentially finds the best approximation solution in one single step if the given affine subspaces are hyperplanes.

References in corpus (5)

• Alternating Projections and Douglas-Rachford for Sparse Affine Feasibility
• Recent Results on Douglas-Rachford Methods for Combinatorial Optimization Problems
• A Cyclic Douglas-Rachford Iteration Scheme
• Circumcentering the Douglas--Rachford method
• On the linear convergence of the circumcentered-reflection method

Cited by in corpus (6)

• On the Circumcentered-Reflection Method for the Convex Feasibility Problem
• The circumcentered-reflection method achieves better rates than alternating projections
• On the centralization of the circumcentered-reflection method
• A successive centralized circumcenter reflection method for the convex feasibility problem
• Circumcentric directions of cones
• A finitely convergent circumcenter method for the Convex Feasibility Problem