Convexification of nonconvex functions and application to minimum and maximum principles for nonconvex sets
type: Item
instance of: scholarly article
title: Convexification of nonconvex functions and application to minimum and maximum principles for nonconvex sets
zbMATH Open document ID: 0873.46013
DOI: 10.1016/0898-1221(96)00016-8
author: Congxin Wu
author: Lixin Cheng
author: Ming-Hu Ha
author: E. Stanley Lee
publication date: 1997-10-27T00:00:00Z
review text: The authors discuss the convexification of nonconvex functions in topological linear spaces. So the well-known results are pointed out that the convex envelope \(C_{nv}(f)\) and the lower semicontinuous convex envelope \(\overline C_{nv}(f)\) of a function \(f: E\to\mathbb{R}\) can be described by the convex hull and the closed convex hull of the epigraph of \(f\). The concave envelope \(C_{nc}(f)\) and the upper semicontinuous concave envelope \(\overline C_{nc}(f)\) can be characterized in the same manner replacing the epigraph by the hypograph. Hence, it is \[ \begin{aligned} C_{nv}(f)(x) & =\inf\{\sum \lambda_if(x_i)\mid\lambda_i\geq 0,\;\sum\lambda_i=1,\;\sum\lambda_ix_i=x\},\\ C_{nc}(f)(x) & =\sup\{\sum \lambda_if(x_i)\mid\lambda_i\geq 0,\;\sum\lambda_i=1,\;\sum\lambda_ix_i= x\}\end{aligned} \] and \[ \begin{aligned} \overline{C}_{nv}(f)(x) &=\inf\{\alpha\mid \alpha=\lim_l\inf C_{nv}(f)(x_i),\;\{x_l\}\to x\},\\ \overline C_{nc}(f)(x) &=\sup\{\alpha\mid\alpha= \lim_l\sup C_{nc}(f)(x_i),\;\{x_l\}\to x\}.\end{aligned} \] The results are used for the proof of different minimum and maximum principles for semicontinuous functions. All the assertions are extensions of the known Stegall theorem to nonconvex sets and have the following form: If \(f\) is a lower/upper semicontinuous and lower/upper bounded function on a set \(A\subseteq E\) (or \(A\subseteq E^*\)) which is closed, bounded and admits an additional property (e.g. \(A\) is contained in a closed convex set with Radon-Nikodým property), then for any \(\varepsilon>0\) there exists a point \(u\in E^*\) \((u\in E)\) with \(|u|<\varepsilon\) such that \(f(\cdot)+ u(\cdot)\) attains a strong minimum/maximum on \(A\).
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 49J50
Mathematics Subject Classification ID: 49J27
zbMATH DE Number: 880527
zbMATH Keywords: convexification
zbMATH Keywords: maximum principle
zbMATH Keywords: Radon-Nikodým property
zbMATH Keywords: nonconvex functions
zbMATH Keywords: lower semicontinuous convex envelope
zbMATH Keywords: Stegall theorem
reviewed by: Jörg Thierfelder
MaRDI profile type: Publication
full work available at URL: 0898-1221(96)00016-8
OpenAlex ID: W1985450790
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published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 880527