Stability results for Ekeland's \(\varepsilon\) variational principle for vector valued functions
type: Item
instance of: scholarly article
title: Stability results for Ekeland's \(\varepsilon\) variational principle for vector valued functions
zbMATH Open document ID: 0941.49011
published in: Mathematical Methods of Operations Research
publication date: 1999-10-05T00:00:00Z
review text: Under the assumption that the nonconvex vector valued function \(f\) satisfies some lower semicontinuity property and is bounded below, the nonconvex vector valued function sequence \(f_n\) satisfies the same lower semicontinuity property and is uniformly bounded below, and \(f_n\) converges to \(f\) in the generalized sense of Mosco, the authors obtain the relation: \[ \sqrt\varepsilon- \text{ext }f= \{\overline x: f(x)- f(\overline x)+ \sqrt\varepsilon\|x-\overline x\|e\not\in -C,\text{ when }x\neq \overline x\}\subseteq \varliminf_{n\to\infty} \sqrt\varepsilon- \text{ext }f_n, \] where \[ \sqrt\varepsilon- \text{ext }f_n= \{\overline x: f_n(x)- f_n(\overline x)+ \sqrt\varepsilon\|x-\overline x\|e\not\in -C,\text{ when }x\neq \overline x\}, \] \(C\) is the pointed closed convex dominating cone with nonempty interior \(\text{int }C\), \(e\in \text{int }C\). Under some conditions, the authors also prove the same result if \(f_n\) converges to \(f\) in the generalized sense of Painlevé-Kuratowski.
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 90C29
Mathematics Subject Classification ID: 90C31
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 1326496
zbMATH Keywords: Mosco convergence
zbMATH Keywords: Painlevé-Kuratowski convergence
zbMATH Keywords: convergence of a sequence of sets
zbMATH Keywords: stability of variational principle for vector valued functions
zbMATH Keywords: lower semicontinuity
author: Guang-Ya Chen
author: Xuexiang Huang
reviewed by: Jürgen Guddat
MaRDI profile type: Publication
full work available at URL: s001860050014
OpenAlex ID: W2012660689
DOI: 10.1007/S001860050014
description: scientific article; zbMATH DE number 1326496