Journal of Applied and Numerical Optimization

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DOI: 10.23952/jano.5.2023.3.02
Volume 5, Issue 3, 1 December 2023, Pages 299-320

 

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 that, in this setting, CRM is globally convergent to a common fixed-point (supposing at least one exists). We also establish the 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.

 

How to Cite this Article:
R. Arefidamghani, R. Behling, A.N. Iusem, L.-R. Santos, A circumcentered-reflection method for finding common fixed points of firmly nonexpansive operators, J. Appl. Numer. Optim. 5 (2023), 299-320.