Mean value theorems and sufficient optimality conditions for nonsmooth functions
type: Item
instance of: scholarly article
title: Mean value theorems and sufficient optimality conditions for nonsmooth functions
zbMATH Open document ID: 0594.49019
DOI: 10.1016/0022-247X(85)90218-5
published in: Journal of Mathematical Analysis and Applications
publication date: 1985-01-01T00:00:00Z
review text: Let X be a locally convex space. Several variants of the mean value theorem for functions \(f: X\to [-\infty,\infty]\) have been proved by \textit{J. B. Hiriart-Urruty} [Numer. Funct. Anal. Optimization 2, 1-30 (1980; Zbl 0462.49032)]. In the paper under review, the first order convex approximation of f introduced by \textit{A. D. Ioffe} [SIAM J. Control Optimization 17, 245-250 (1979; Zbl 0417.49027)] is modified to be applicable to the case when both f and its approximation may take on \(+\infty\) and -\(\infty\). Then variants of the mean value theorems are obtained in terms of subdifferentials of these modified approximations. An application to the problem of minimization of non-differentiable not necessarily convex functions on a convex subset of X is presented.
Mathematics Subject Classification ID: 49K27
Mathematics Subject Classification ID: 26E15
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 46A03
zbMATH DE Number: 3955834
zbMATH Keywords: nonsmooth analysis
zbMATH Keywords: optimality conditions
zbMATH Keywords: mean value theorem
zbMATH Keywords: first order convex approximation
zbMATH Keywords: subdifferentials
author: Marcin Studniarski
reviewed by: Milan Vlach
MaRDI profile type: Publication
cites work: On Sufficient Conditions in Nonsmooth Optimization
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cites work: Mean value theorem for functions possessing first order convex approximations. Applications in optimization theory
cites work: A SIMPLE DERIVATION OF SUFFICIENT CONDITIONS FOR A LOCAL MINIMUM OF A LIPSCHITZIAN FUNCTION
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description: scientific article; zbMATH DE number 3955834