Regularity and stability for a convex feasibility problem

TL;DR

This paper investigates the stability of iterative projections onto perturbed convex sets, establishing conditions under which the sequences converge to the intersection, even with set perturbations.

Contribution

It proves that bounded regularity of convex set pairs ensures stability of projection sequences under set perturbations.

Findings

Projection sequences converge to the intersection under regularity.

Stability holds even with set perturbations.

Results extend stability analysis in convex feasibility problems.

Abstract

Let us consider two sequences of closed convex sets $\{A_n\}$ and $\{B_n\}$ converging with respect to the Attouch-Wets convergence to $A$ and $B$, respectively. Given a starting point $a_0$, we consider the sequences of points obtained by projecting on the "perturbed" sets, i.e., the sequences $\{a_n\}$ and $\{b_n\}$ defined inductively by $b_n=P_{B_n}(a_{n-1})$ and $a_n=P_{A_n}(b_n)$. Suppose that $A\cap B$ (or a suitable substitute if $A \cap B=\emptyset$) is bounded, we prove that if the couple $(A,B)$ is (boundedly) regular then the couple $(A,B)$ is $d$-stable, i.e., for each $\{a_n\}$ and $\{b_n\}$ as above we have $\mathrm{dist}(a_n,A\cap B)\to 0$ and $\mathrm{dist}(b_n,A\cap B)\to 0$.