A circumcentered-reflection method for finding common fixed points of firmly nonexpansive operators

TL;DR

This paper extends the circumcentered-reflection method (CRM) to find common fixed points of firmly nonexpansive operators, proving its global and linear convergence, and demonstrating its superiority over classical methods through numerical experiments.

Contribution

The paper generalizes CRM to a broader class of operators, establishes convergence properties, and compares its performance favorably against traditional projection methods.

Findings

CRM is globally convergent for firmly nonexpansive operators.

CRM exhibits linear convergence under an error bound condition.

Numerical results show CRM outperforms the classical Parallel Projections Method.

Abstract

The circumcentered-reflection method (CRM) has been recently proposed as a methodology for accelerating several algorithms for solving the Convex Feasibility Problem (CFP), equivalent to finding a common fixed-point of the orthogonal projections onto a finite number of closed and convex sets. In this paper, we apply CRM to the more general Fixed Point Problem (denoted as FPP), consisting of finding a common fixed-point of operators belonging to a larger family of operators, namely firmly nonexpansive operators. We prove than in this setting, CRM is globally convergent to a common fixed-point (supposing at least one exists). We also establish linear convergence of the sequence generated by CRM applied to FPP, under a not too demanding error bound assumption, and provide an estimate of the asymptotic constant. We provide solid numerical evidence of the superiority of CRM when compared to the classical Parallel Projections Method (PPM). Additionally, we present certain results of convex combination of orthogonal projections, of some interest on its own.