Subdifferential characterization of approximate convexity: The lower semicontinuous case
type: Item
instance of: scholarly article
title: Subdifferential characterization of approximate convexity: The lower semicontinuous case
zbMATH Open document ID: 1160.26007
author: A. Daniilidis
author: Florence Jules
author: Marc Lassonde
published in: Mathematical Programming. Series A. Series B
publication date: 2008-12-16T00:00:00Z
review text: Having a Banach space \(X\) and a lower semicontinuous function \(f:X\to \mathbb{R}\cup\{+\infty\}\), the authors show as the main result of the paper that \(f\) is (directionally) approximately convex at \(x_0\in X\) if and only if \(\partial_Cf\) is (directionally) sub\-monotone at \(x_0\), where \(\partial_Cf\) denotes the Clarke subdifferential of \(f\). For the proof, the authors use a so called three points approximate mean value inequality stated and proved in the paper. The case of arbitrary subdifferentials is also discussed.
Mathematics Subject Classification ID: 26B25
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 5382670
zbMATH Keywords: approximate convexity
zbMATH Keywords: submonotone operator
zbMATH Keywords: subdifferential
zbMATH Keywords: mean value inequality
reviewed by: Constantin Zălinescu
MaRDI profile type: Publication
full work available at URL: s10107-007-0127-3
OpenAlex ID: W1977663139
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DOI: 10.1007/S10107-007-0127-3
description: scientific article; zbMATH DE number 5382670