Prox-regular functions in Hilbert spaces
type: Item
instance of: scholarly article
title: Prox-regular functions in Hilbert spaces
zbMATH Open document ID: 1085.49020
author: Frédéric Bernard
author: L. Thibault
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-03-08T00:00:00Z
review text: The authors extend results about prox-regular functions known for finite-dimensional spaces to infinite-dimensional Hilbert spaces. After the introduction of prox-regularity of a function \(f\) by means of the limiting subdifferential \(\partial f\) it is pointed out that convex functions, so-called convexly \(C^{1,+}\)-composite functions and primal lower nice functions (in sense of Poliquin) are proxregular. With the help of the Moreau envelopes for \(f\) and the associated proximal mappings, a subdifferential characterization of prox-regularity is given showing that certain localizations of the subdifferential (the so-called \(f\)-attentive \(\varepsilon\)-localization of \(\partial f\)) admit a monotonicity behavior. Moreover, links between these locations and the proximal mappings of \(f\) are pointed out. At the end of the paper, \(C^{1,+}\)-regularity properties of the Moreau envelopes and Lipschitz properties of the proximal mappings of prox-regular functions are presented.
reviewed by: Jörg Thierfelder
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 26E15
Mathematics Subject Classification ID: 58C20
zbMATH DE Number: 2141737
zbMATH Keywords: prox-regular function
zbMATH Keywords: subdifferential
zbMATH Keywords: Moreau envelope
zbMATH Keywords: proximal mapping
zbMATH Keywords: primal lower nice function
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.06.003
OpenAlex ID: W2016104348
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DOI: 10.1016/J.JMAA.2004.06.003
description: scientific article; zbMATH DE number 2141737