Proximal analysis in smooth spaces
type: Item
instance of: scholarly article
title: Proximal analysis in smooth spaces
zbMATH Open document ID: 0858.49015
DOI: 10.1007/BF00419371
published in: Set-Valued Analysis
publication date: 1997-03-19T00:00:00Z
review text: The authors discuss different strategies for the definition of subdifferentials for lower semicontinuous functions on a Banach space \(X\). Many notions -- especially the Clarke subdifferential and the approximate \(G\)-subdifferential of Ioffe -- are introduced by a topological way in three steps: (i) definition of \(\partial f\) for Lipschitz functions, (ii) definition of normal cones by the subdifferential of the distance function, (iii) definition of \(\partial f\) for a general function \(f\) via normal cone to the epigraph. It is the aim of the paper to give a sequential description for such subdifferentials, i.e., a description of these subdifferentials by limits of ``classical'' differentiability notions. Here, with classical differentiability the authors mean \(\beta\)-differentiability and \(\beta\)-subdifferentiability with respect to an arbitrary bornology \(\beta\). Clearly, using the bornology of all bounded, compact or finite subsets, we get the differentiability notions in the sense of Fréchet, Hadamard and Gâteaux, respectively. The results are summarized in three main theorems. Assuming a \(\beta\)-smooth norm in \(X\), Theorem 1 gives a representation of the \(G\)-normal cone (Ioffe) and the \(C\)-normal cone (Clarke) of a set as special limits of \(\beta\)-normal cones. An analogous description for the \(G\)-subdifferential and the Clarke subdifferential of a lower semicontinuous function as limit of \(\beta\)-subdifferentials is formulated in Theorem 2. Finally, Theorem 3 concerns to the singular \(G\)-subdifferential and the singular Clarke subdifferential in the Fréchet case, i.e., using the bornology of bounded sets.
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 49J50
Mathematics Subject Classification ID: 58C20
Mathematics Subject Classification ID: 26B05
zbMATH DE Number: 898037
zbMATH Keywords: \(\beta\)-derivative
zbMATH Keywords: lower semicontinuous functions
zbMATH Keywords: Clarke subdifferential
zbMATH Keywords: approximate \(G\)-subdifferential of Ioffe
zbMATH Keywords: Lipschitz functions
zbMATH Keywords: bornology
zbMATH Keywords: \(\beta\)-subdifferentials
author: Jonathan M. Borwein
author: Alexander D. Ioffe
reviewed by: Jörg Thierfelder
MaRDI profile type: Publication
cites work: Q3976070
cites work: A Weak Hadamard Smooth Renorming of L1(Ω, μ)
cites work: Q4856858
cites work: Minimal Convex Uscos and Monotone Operators on Small Sets
cites work: Proximal Analysis and Boundaries of Closed Sets in Banach Space, Part I: Theory
cites work: Proximal Analysis and Boundaries of Closed Sets in Banach Space. Part II: Applications
cites work: Q4338513
cites work: On the variational principle
cites work: Calculus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps
cites work: Approximate subdifferentials and applications 3: the metric theory
cites work: Proximal Analysis and Approximate Subdifferentials
cites work: Generalized differentials of nonsmooth functions, and necessary conditions for an extremum
cites work: Gateaux Differentiability of Convex Functions on Banach Spaces
cites work: Nonsmooth sequential analysis in Asplund spaces
cites work: Extremal characterizations of asplund spaces
cites work: Differentiability of Lipschitz functions on Banach spaces
cites work: Clarke's Gradients and Epsilon-Subgradients in Banach Spaces
full work available at URL: https://doi.org/10.1007/bf00419371
OpenAlex ID: W1980034892
description: scientific article; zbMATH DE number 898037