SIP: critical value functions have finite modulus of non-convexity
type: Item
instance of: scholarly article
title: SIP: critical value functions have finite modulus of non-convexity
zbMATH Open document ID: 1270.90086
author: Hubertus T. Jongen
author: Dominik Dorsch
author: Francisco Guerra Vazquez
author: Harald Günzel
author: Jan-Joachim Rückmann
published in: Mathematical Programming. Series A. Series B
publication date: 2012-12-19T00:00:00Z
review text: The authors consider semi-infinite programming problems \(SIP(z)\) depending on a finite dimensional parameter \(z \in \mathbb{R}^p\). In this work, the authors generalize the results obtained by \textit{H. Günzel} et al. [SIAM J. Optim. 16, No. 4, 1044--1053 (2006; Zbl 1131.90044)] to the case of critical value functions of parametric semi-infinite programs (SIP). As in [Zbl 1131.90044], the techniques used in order to deduce their results, strongly depend on the characterization of strong stability of stationary points. In the abstract it is said: ``Provided that \(\bar x\) is a strongly stable stationary point of \(SIP(\bar z)\), there exists a locally unique and continuous stationary point mapping \(z\rightarrow x(z)\). This defines the local critical value function \(\varphi(z)@>\text{def}>>{=}f(x(z);z)\), where \(x\rightarrow f(x;z)\) denotes the objective function of \(SIP(z)\) for a given parameter vector \(z \in \mathbb{R}^p\). We show that \(\varphi\) is the sum of a convex function and a smooth function. In particular, this excludes the appearance of negative kinks in the graph of \(\varphi\).''
Mathematics Subject Classification ID: 90C34
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90C31
zbMATH DE Number: 6116931
zbMATH Keywords: SIP
zbMATH Keywords: semi-infinite programming
zbMATH Keywords: parametric optimization
zbMATH Keywords: strong stability
zbMATH Keywords: modulus of nonconvexity
zbMATH Keywords: critical value function
reviewed by: Nada Djuranović-Miličić
MaRDI profile type: Publication
full work available at URL: s10107-012-0554-7
OpenAlex ID: W2040432381
cites work: Q4903672
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DOI: 10.1007/S10107-012-0554-7
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description: scientific article; zbMATH DE number 6116931