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 $Acap B$.
Lingua originaleEnglish
pagine (da-a)N/A-N/A
RivistaSet-Valued and Variational Analysis
DOI
Stato di pubblicazionePubblicato - 2021

Keywords

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

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