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When some variational properties force convexity

Published online by Cambridge University Press:  17 May 2013

M. Volle ,
J.-B. Hiriart-Urruty  and
C. Zălinescu
M. Volle
Affiliation:
Department of Mathematics, University of Avignon, France. michel.volle@univ-avignon.fr
J.-B. Hiriart-Urruty
Affiliation:
Institut de Mathématiques, Université Paul Sabatier, Toulouse, France; jbhu@cict.fr
C. Zălinescu
Affiliation:
Faculty of Mathematics, University Al. I. Cuza, Iaşi, Romania Octav Mayer Institute of Mathematics, Romanian Academy, Romania; zalinesc@uaic.ro
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Abstract

The notion of adequate (resp. strongly adequate) function has been recently introduced to characterize the essentially strictly convex (resp. essentially firmly subdifferentiable) functions among the weakly lower semicontinuous (resp. lower semicontinuous) ones. In this paper we provide various necessary and sufficient conditions in order that the lower semicontinuous hull of an extended real-valued function on a reflexive Banach space is essentially strictly convex. Some new results on nearest (farthest) points are derived from this approach.

Keywords

Convex dualitywell posed optimization problemessential strict convexityessential smoothnessbest approximation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 701 - 709
Copyright
© EDP Sciences, SMAI, 2013

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