Characterization of regularity via variational stability of alternating projections sequences
Optimization and Control
2026-05-05 v2
Abstract
The notion of regular pair for two nonempty closed convex subsets and~ of a Hilbert space \H was introduced by Borwein and Bauschke in 1993 to ensure convergence (in norm) of the alternating projection method to some point of the best approximation set. In 2022, De Bernardi and Miglierina showed that regularity of the pair guarantees, additionally, the convergence for any variational perturbation of the alternating projection method, provided the corresponding best approximation sets are bounded. In this work, we show that the converse assertion is also true. Moreover, this converse assertion holds without requiring the best approximation sets to be bounded.
Cite
@article{arxiv.2511.07953,
title = {Characterization of regularity via variational stability of alternating projections sequences},
author = {Francesco Battistoni and Aris Daniilidis and Carlo Alberto De Bernardi and Enrico Miglierina},
journal= {arXiv preprint arXiv:2511.07953},
year = {2026}
}