Stability of a convex feasibility problem

TL;DR

This paper investigates the stability of solutions to convex feasibility problems under set perturbations, ensuring convergence of solutions in both finite and infinite-dimensional spaces with various examples.

Contribution

It establishes stability results for convex feasibility problems with converging set sequences, extending to infinite-dimensional spaces and illustrating the importance of assumptions.

Findings

Solutions of perturbed problems converge to original solutions

Stability results hold in both finite and infinite-dimensional spaces

Examples highlight the necessity of assumptions for convergence

Abstract

The 2-sets convex feasibility problem aims at finding a point in the intersection of two closed convex sets $A$ and $B$ in a normed space $X$. More generally, we can consider the problem of finding (if possible) two points in $A$ and $B$, respectively, which minimize the distance between the sets. In the present paper, we study some stability properties for the convex feasibility problem: we consider two sequences of sets, each of them converging, with respect to a suitable notion of set convergence, respectively, to $A$ and $B$. Under appropriate assumptions on the original problem, we ensure that the solutions of the perturbed problems converge to a solution of the original problem. We consider both the finite-dimensional and the infinite-dimensional case. Moreover, we provide several examples that point out the role of our assumptions in the obtained results.