Finite Convergence of Locally Proper Circumcentered Methods

In view of the great performance of circumcentered isometry methods for solving the best approximation problem, in this work we further investigate the locally proper circumcenter mapping and circumcentered method. Various examples of locally proper circumcenter mapping are presented and studied. Inspired by some results on the circumcentered-reflection method by R. Arefidamghani, R. Behling, Y. Bello-Cruz, A. N. Iusem and L. R. Santos [The block-wise circumcentered-reflection method, Comp. Optimization Appl. 76/3 (2019) 675--699; The circumcentered-reflection method achieves better rates than alternating projections, ibid. 79/2 (2021) 507--530; On the circumcentered-reflection method for the convex feasibility problem, Num. Algorithms 86/4 (2021) 1475--1494], we provide sufficient conditions for one-step convergence of circumcentered isometry methods for finding the best approximation point onto the intersection of fixed point sets of related isometries. In addition, we elaborate the performance of circumcentered reflection methods induced by reflectors associated with hyperplanes and halfspaces for finding the best approximation point onto (or a point in) the intersection of hyperplanes and halfspaces.