Extensions of Fréchet \(\varepsilon\)-subdifferential calculus and applications
type: Item
instance of: scholarly article
title: Extensions of Fréchet \(\varepsilon\)-subdifferential calculus and applications
zbMATH Open document ID: 1029.49015
author: Huynh Van Ngai
author: Dinh The Luc
author: Michel Théra
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-07-04T00:00:00Z
review text: The authors present calculus rules using the limiting Fréchet \(\varepsilon\)-subdifferentials of lower semicontinuous functions and the limiting Fréchet \(\varepsilon\)-normals of closed sets in Banach spaces, especially in Asplund spaces. In the first part of the paper, a calculus formula for the limiting Fréchet \(\varepsilon\)-subdifferential \(\widehat\partial_\varepsilon p(\overline u)\) of the marginal function \[ p(u)= \min_{x\in F(u)} \varphi(u, x) \] (where \(\varphi: U\times X\to \mathbb{R}\cup\{+\infty\}\) is a lower semicontinuous function and \(F: U\rightrightarrows X\) is a set-valued mapping with closed graph) is given in terms of the limiting Fréchet \(\varepsilon\)-subdifferential \(\widehat\partial_\varepsilon \varphi(\overline u,\overline x)\) and the limiting Fréchet \(\varepsilon\)-normal \(\widehat N(\text{graph }F,\cdot)(\overline u,\overline x)\). In the second part, the authors present a calculus rule for the limiting Fréchet \(\varepsilon\)-subdifferential \(\widehat\partial_\varepsilon(g\circ F)(\overline x)\) of the composite function \(g\circ F: X\to \mathbb{R}\cup \{+\infty\}\) (where \(F: X\to Y\) is a strictly (compactly) Lipschitzian mapping and \(g: Y\to \mathbb{R}\cup\{+\infty\}\) is a lower semicontinuous function) in terms of the limiting Fréchet \(\varepsilon\)-subdifferential \(\widehat\partial_\varepsilon(y^*\circ F)(\overline x)\) where \(y^*\) is contained in the limiting Fréchet subdifferential \(\widehat\partial g(F(\overline x))\). In the last part of the paper, the results are used to derive necessary optimality conditions for the approximate solutions of the general nonsmooth constraint optimization problem \[ \min f(x)\qquad\text{s.t. }F(x)\in -S, \] where \(f: X\to \mathbb{R}\) is a Lipschitz function, \(F: X\to Y\) is a strictly compactly Lipschitzian mapping and \(S\subset Y\) is a nonempty closed convex cone.
reviewed by: Jörg Thierfelder
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 26E15
zbMATH DE Number: 1763457
zbMATH Keywords: calculus rules
zbMATH Keywords: limiting Fréchet \(\varepsilon\)-subdifferentials
zbMATH Keywords: Banach spaces
zbMATH Keywords: Asplund spaces
zbMATH Keywords: marginal function
zbMATH Keywords: composite function
zbMATH Keywords: necessary optimality conditions
Wikidata QID: Q57580565
MaRDI profile type: Publication
full work available at URL: jmaa.2001.7832
OpenAlex ID: W1974018779
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DOI: 10.1006/JMAA.2001.7832
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description: scientific article; zbMATH DE number 1763457