Are the approximate and the Clarke subgradients generically equal?
type: Item
instance of: scholarly article
title: Are the approximate and the Clarke subgradients generically equal?
zbMATH Open document ID: 0836.49010
author: Guy Katriel
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-04-29T00:00:00Z
review text: Two of the well-known subdifferential notions for a real-valued function \(f : \mathbb{R}^n \to \mathbb{R}\) -- the Clarke subdifferential \(\partial_c f(x)\) and the approximate subdifferential \(\partial_a f(x)\) of Joffe - - are compared in the present paper. Starting with the formerly proved inclusion \[ \partial_a f(x) \subseteq \partial_c f(x) \] it is observed if both sets are generically equal, either in sense of measure theory (i.e., equal on a set of full measure) or in the topological sense (i.e., equal on a dense \(G_\delta\)-set). In the one-dimensional case, positive answers are given showing that both subdifferentials are different at most on a countable set. For higher- dimensional spaces, however, the problem is more complicated. The author constructs a function for which the subdifferentials are different on a set with positive measure. In this example, the set on which the subdifferentials are equal is of positive measure, too, and a dense \(G_\delta\)-set. Hence, the question for the generic equality in topological sense remains unanswered.
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 26A27
Mathematics Subject Classification ID: 26E25
zbMATH DE Number: 811192
zbMATH Keywords: approximate subdifferential of Joffe
zbMATH Keywords: Clarke subdifferential
zbMATH Keywords: generic equality
reviewed by: Jörg Thierfelder
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1255
OpenAlex ID: W1964926425
DOI: 10.1006/JMAA.1995.1255
description: scientific article; zbMATH DE number 811192