Characterization of regularity via variational stability of alternating projections sequences

TL;DR

This work addresses when variational perturbations of the alternating projections converge by linking two classical notions: regularity of a pair of convex sets and $d$-stability. Under the assumption that the best-approximation sets $E$ and $F$ are nonempty and bounded, the authors prove that $d$-stability implies regularity, complementing existing results that regularity implies $d$-stability, thereby establishing Theorem A. The proof combines a two-dimensional reduction, a perturbation technique to create nearby convex sets, and Attouch-Wets convergence arguments to transfer stability to the original problem. The result yields a complete characterization of regularity via variational stability for the convex two-set feasibility problem, clarifying when perturbed alternating projection sequences converge in norm and informing algorithmic design.

Abstract

The notion of regular pair $(A,B)$ for two nonempty closed convex subsets $A$ and~$B$ of a Hilbert space $\mathcal{H}$ 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. The aim of this paper is to show that the converse assertion is also true.

Theorems & Definitions (19)

• Definition 1.1: $d$-stability
• Theorem 1
• Definition 2.1: Hausdorff convergence
• Definition 2.2: Attouch-Wets convergence
• proof
• Definition 2.6: regular and boundedly regular pair
• Proposition 2.7
• Theorem 2.8
• Lemma 3.1: keeping iterations in a 2-dim space
• proof