TY - JOUR

T1 - Regularity and Stability for a Convex Feasibility Problem

AU - De Bernardi, Carlo Alberto

AU - Miglierina, Enrico

PY - 2021

Y1 - 2021

N2 - 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 $Acap B$.

AB - 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 $Acap B$.

KW - Alternating projections method

KW - Convex feasibility problem

KW - Regularity

KW - Set-convergence

KW - Stability

KW - Alternating projections method

KW - Convex feasibility problem

KW - Regularity

KW - Set-convergence

KW - Stability

UR - http://hdl.handle.net/10807/183435

U2 - 10.1007/s11228-021-00602-3

DO - 10.1007/s11228-021-00602-3

M3 - Article

SP - N/A-N/A

JO - Set-Valued and Variational Analysis

JF - Set-Valued and Variational Analysis

SN - 1877-0533

ER -