Porosity of perturbed optimization problems in Banach spaces
type: Item
instance of: scholarly article
title: Porosity of perturbed optimization problems in Banach spaces
zbMATH Open document ID: 1101.49007
author: C. Li
author: Lihui Peng
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-11-15T00:00:00Z
review text: Let \(X\) be a Banach space and \(Z\) a nonempty closed subset of \(X\). Let \(J:Z\to\mathbb R\) be a lower semicontinuous function bounded from below. This paper is concerned with the perturbed optimization problem \(\inf_{z\in Z}\{J(z)+\|x-z\|\}\), denoted by \((x,J)\)-inf for \(x\in X\). In the case when \(X\) is compactly fully 2-convex, it is proved in the present paper that the set of all points \(x\) in \(X\) for which there does not exist \(z_0\in Z\) such that \(J(z_0)+ \|x-z_0\|= \inf_{z\in Z}\{J(z0+ \|x-z\|\}\) is a \(\sigma\)-porous set in \(X\). Furthermore, if \(X\) is assumed additionally to be compactly locally uniformly convex, we verify that the set of all points \(x\in X\setminus Z^0\) such that the problem \((x,J)\)-inf fails to be approximately compact, is a \(\sigma\)-porous set in \(X\setminus Z^0\), where \(Z^0\) denotes the set of all \(z\in Z\) such that \(z\in P_Z(z)\). Moreover, a counterexample to which some results of \textit{R. X. Ni} [J. Math. Anal. Appl. 302, No. 2, 417--424 (2005; Zbl 1061.49002)] fail is provided.
Mathematics Subject Classification ID: 49J27
zbMATH DE Number: 5072981
zbMATH Keywords: \(\sigma\)-porous set
zbMATH Keywords: perturbed optimization problem
zbMATH Keywords: lower semicontinuous function
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.12.030
OpenAlex ID: W2000106857
cites work: Q5638791
cites work: Existence de solutions pour des problèmes d'optimisation non convexe
cites work: Existence theorems for usual and approximate solutions of optimal control problems
cites work: On a generalized best approximation problem
cites work: Porous Sets in Best Approximation Theory
cites work: Q4043661
cites work: Nonconvex Optimization Problems Depending on a Parameter
cites work: Q3906015
cites work: Generic existence of solutions for some perturbed optimization problems
cites work: Q3213662
cites work: Well-posed optimization problems
cites work: FULLY CONVEX NORMED LINEAR SPACES
cites work: Nearest points to closed sets and directional derivatives of distance functions
cites work: Q3208460
cites work: On well posedness of best simultaneous approximation problems in Banach spaces.
cites work: On generic well-posedness of restricted Chebyshev center problems in Banach spaces
cites work: Porosity of mutually nearest and mutually furthest points in Banach spaces.
cites work: On the k-uniform rotund and the fully convex Banach spaces
cites work: An Introduction to Banach Space Theory
cites work: Q5644664
cites work: Generic solutions for some perturbed optimization problem in non-reflexive Banach spaces
cites work: Q5596489
cites work: Porosity and \(\sigma\)-porosity
DOI: 10.1016/J.JMAA.2005.12.030
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description: scientific article; zbMATH DE number 5072981