Characterization of regularity via variational stability of alternating projections sequences

TL;DR

This paper establishes a bi-conditional relationship between regularity of convex pairs in Hilbert spaces and the stability of alternating projections under variational perturbations, extending previous results.

Contribution

It proves that the variational stability of alternating projections characterizes regular pairs without the boundedness assumption of best approximation sets.

Findings

Regularity of convex pairs is equivalent to variational stability of alternating projections.

The equivalence holds even without boundedness of the best approximation sets.

The results extend previous work by removing the boundedness condition.

Abstract

The notion of regular pair $(A,B)$ for two nonempty closed convex subsets $A$ and~$B$ 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 $(A,B)$ 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.