Characterization of strict convexity for locally Lipschitz functions
type: Item
instance of: scholarly article
title: Characterization of strict convexity for locally Lipschitz functions
zbMATH Open document ID: 1192.49019
DOI: 10.1216/RMJ-2009-39-6-2029
author: Karel Pastor
published in: Rocky Mountain Journal of Mathematics
publication date: 2010-01-05T00:00:00Z
review text: It is well known that a smooth function \(f: X\to\mathbb{R}\) is convex iff \(f''(x;h, h)\geq 0\) for all \(x\in X\) and all \(h\in X\). The function \(f\) is strictly convex iff \(f''(x;h,h)> 0\) for ``almost'' all \(x\in X\) and all \(h\in X\). Here \[ f''(x;u,v)= \lim_{t\to 0} \frac{\langle\nabla f(x+ tu)-\nabla f(x),v\rangle}{t}, \] where \(\nabla f(x)\) is the symbol for the Gâteaux derivative of \(f\) at \(x\). In the paper, the author extends this result for nonsmooth, but locally Lipschitz functions using generalized directional derivatives of second order in which the gradient \(\nabla f\) is replaced by the Clarke subdifferential mapping \(\partial_C f\) and the limit is replaced by generalized limits. After the characterization of convexity for regularly locally Lipschitz functions, sufficient and necessary conditions for monotonicity and strict monotonicity of set-valued mappings \(F: X\rightrightarrows X^*\) are provided. Setting \(F=\partial_C f\), the author presents the main theorem of the paper using the fact that a real-valued function \(f\) is (strictly) convex if and only if the mapping \(\partial_C f\) is (strictly) monotone.
reviewed by: Jörg Thierfelder
Mathematics Subject Classification ID: 49J52
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 52A41
Mathematics Subject Classification ID: 58C05
Mathematics Subject Classification ID: 58C06
Mathematics Subject Classification ID: 58C20
Mathematics Subject Classification ID: 26B25
zbMATH DE Number: 5652736
zbMATH Keywords: strict convexity
zbMATH Keywords: generalized second-order derivative
zbMATH Keywords: locally Lipschitz functions
zbMATH Keywords: monotone operator
zbMATH Keywords: Clarke subdifferential
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5652736