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 $Acap B$.

De Bernardi, C. A., Miglierina, E., Regularity and Stability for a Convex Feasibility Problem, <<SET-VALUED AND VARIATIONAL ANALYSIS>>, 2022; 30 (2): 521-542. [doi:10.1007/s11228-021-00602-3] [https://hdl.handle.net/10807/183435]

Regularity and Stability for a Convex Feasibility Problem

De Bernardi, Carlo Alberto
;
Miglierina, Enrico
2021

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 $Acap B$.
2021
Inglese
De Bernardi, C. A., Miglierina, E., Regularity and Stability for a Convex Feasibility Problem, <<SET-VALUED AND VARIATIONAL ANALYSIS>>, 2022; 30 (2): 521-542. [doi:10.1007/s11228-021-00602-3] [https://hdl.handle.net/10807/183435]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/183435
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