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 onto 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 $Acap B$ 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 $mathrm{dist}(a_n,Acap B) o 0$ and $mathrm{dist}(b_n,Acap B) o 0$. Similar results are obtained also in the case $A cap B=emptyset$, considering the set of best approximation pairs instead of $Acap B$.
Original languageEnglish
Pages (from-to)N/A-N/A
JournalSet-Valued and Variational Analysis
DOIs
Publication statusPublished - 2021

Keywords

  • Alternating projections method
  • Convex feasibility problem
  • Regularity
  • Set-convergence
  • Stability

Fingerprint

Dive into the research topics of 'Regularity and Stability for a Convex Feasibility Problem'. Together they form a unique fingerprint.

Cite this