Links between directional derivatives through multidirectional mean value inequalities
type: Item
instance of: scholarly article
title: Links between directional derivatives through multidirectional mean value inequalities
zbMATH Open document ID: 1160.26004
author: Rafael Correa
author: P. Gajardo
author: L. Thibault
published in: Mathematical Programming. Series A. Series B
publication date: 2008-12-16T00:00:00Z
full work available at URL: http://hdl.handle.net/10533/127844
review text: The aim of the article is to extend certain inequalities between Rockafellar's directional derivatives and the mixed lower limit of the lower Dini's derivatives to the general case of lower semi-continuous (lsc) functions on Banach spaces and also the inclusions between the corresponding tangent cones to closed subsets of Banach spaces. To this end, the authors prove first the so called ``multidirectional mean value inequality'' \[ d^\uparrow f(\overline x;\overline v)\leq \lim \sup_{x\to_f\overline x}\inf_{v\to \overline v} d^-f(x;v) \] between the Rockafellar's directional derivative, \(d^\uparrow f(.;.)\), and the lower Dini's derivatives, \(d^-f(.;.)\), and a similar inequality for the weak lower Dini's derivatives, \(d_w^-f(.;.)\); next, as an easy corollary, one obtains in this context the known inclusion \[ \liminf_{x\to_f\overline x}T_B(S;x)\subset T_C(S;\overline x) \] between Bouligand (contingent) cones and Clarke's tangent cones to a closed subset of a Banach space. In the last part of the paper one studies several classes of functions for which the mean value equality holds.
reviewed by: Ştefan Mirică
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 26A16
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 5382668
zbMATH Keywords: directional derivatives
zbMATH Keywords: tangent cones
zbMATH Keywords: multidirectional mean value inequality
zbMATH Keywords: compactly epi-lipschizian function
MaRDI profile type: Publication
OpenAlex ID: W2022110970
cites work: Q5186236
cites work: Nonsmooth Constrained Optimization and Multidirectional Mean Value Inequalities
cites work: Q5528052
cites work: Epi-Lipschitz-like sets in Banach space: theorems and examples
cites work: Tangential approximations
cites work: Proximal Analysis and Boundaries of Closed Sets in Banach Space, Part I: Theory
cites work: Mean Value Inequalities in Hilbert Space
cites work: Q4375488
cites work: Subdifferential representation formula and subdifferential criteria for the behavior of nonsmooth functions
cites work: Q4257415
cites work: An introduction to \(\Gamma\)-convergence
cites work: Approximate Subdifferentials and Applications. I: The Finite Dimensional Theory
cites work: The role of locally compact cones in nonsmooth analysis.
cites work: Extensions Of Subdifferential Calculus Rules in Banach Spaces
cites work: A strong mean value theorem and applications
cites work: A characterization of tangential regularity
cites work: Directionally Lipschitzian Functions and Subdifferential Calculus
cites work: Generalized Directional Derivatives and Subgradients of Nonconvex Functions
cites work: Variational Analysis
cites work: Integration of subdifferentials of lower semicontinuous functions on Banach spaces
cites work: Characterization of Clarke's tangent and normal cones in finite and infinite dimensions
DOI: 10.1007/S10107-007-0131-7
description: scientific article; zbMATH DE number 5382668