Scalarization of tangential regularity of set-valued mappings
type: Item
instance of: scholarly article
title: Scalarization of tangential regularity of set-valued mappings
zbMATH Open document ID: 0942.49017
DOI: 10.1023/A:1008710527643
published in: Set-Valued Analysis
publication date: 2000-08-17T00:00:00Z
review text: Having a set-valued mapping \(M:E\rightarrow F\) from the Hausdorff topological vector space \(E\) into the normed space \(F\), one associates the scalar mapping \(\Delta _{M}:E\times F\rightarrow \overline{{R}},\) \(\Delta _{M}(x,y)=d(y,M(x)).\) The authors are mainly interested by the relationships between the tangential regularity of \(M\) at \((\overline{x},\overline{y})\in\text{gph} M\) (i.e., the coincidence of Clarke and Bouligand tangent cones of \(\text{gph} M\) at \((\overline{x},\overline{y})\)) and the directional regularity of \(\Delta _{M}\) at \((\overline{x},\overline{y})\) (i.e., the coincidence of Rockafellar and lower Hadamard directional derivatives of \(\Delta _{M}\) at \((\overline{x},\overline{y})\)). It is shown that \(\Delta _{M} \) is directionally regular at \((\overline{x},\overline{y})\) when \(M\) is tangentially regular at \((\overline{x},\overline{y}),\) the converse being true if \(M\) is in the class \({\mathcal{K}}(\overline{x},\overline{y}),\) in particular if \(F\) is finite dimensional. Many other interesting results are also stated.
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 49J53
Mathematics Subject Classification ID: 58C20
Mathematics Subject Classification ID: 90C26
Mathematics Subject Classification ID: 58C06
zbMATH DE Number: 1331786
zbMATH Keywords: set-valued mapping
zbMATH Keywords: tangent cone
zbMATH Keywords: tangential regularity
zbMATH Keywords: directional regularity
zbMATH Keywords: directional derivatives
author: Messaoud Bounkhel
author: L. Thibault
reviewed by: Constantin Zălinescu
MaRDI profile type: Publication
full work available at URL: a:1008710527643
OpenAlex ID: W2117917530
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description: scientific article; zbMATH DE number 1331786