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<datestamp>2018-01-11</datestamp>
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<title lang=en>Mean value property and subdifferential criteria for lower semicontinuous functions</title>
<creator>Aussel, Didier</creator>
<creator>Corvellec, Jean-Noël</creator>
<creator>Lassonde, Marc</creator>
<contributor>Procédés, Matériaux et Energie Solaire (PROMES) ; Université de Perpignan Via Domitia (UPVD) - Centre National de la Recherche Scientifique (CNRS)</contributor>
<contributor>LAboratoire de Mathématiques et PhySique (LAMPS) ; Université de Perpignan Via Domitia (UPVD)</contributor>
<contributor>Laboratoire de Mathématiques Informatique et Applications (LAMIA) ; Université des Antilles et de la Guyane (UAG)</contributor>
<description>International audience</description>
<source>ISSN: 0002-9947</source>
<source>Transactions of the American Mathematical Society</source>
<publisher>American Mathematical Society</publisher>
<identifier>hal-00699220</identifier>
<identifier>https://hal.archives-ouvertes.fr/hal-00699220</identifier>
<source>https://hal.archives-ouvertes.fr/hal-00699220</source>
<source>Transactions of the American Mathematical Society, American Mathematical Society, 1995, 347 (10), pp.4147-4161</source>
<language>en</language>
<subject>[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]</subject>
<type>info:eu-repo/semantics/article</type>
<type>Journal articles</type>
<description lang=en>We define an abstract notion of subdifferential operator and an associated notion of smoothness of a norm covering all the standard situations. In particular, a norm is smooth for the Gâteaux (Fréchet, Hadamard, Lipschitz-smooth) subdifferential if it is Gâteaux (Fréchet, Hadamard, Lipschitz) smooth in the classical sense, while on the other hand any norm is smooth for the Clarke-Rockafellar subdifferential. We then show that lower semicontinuous functions on a Banach space satisfy an Approximate Mean Value Inequality with respect to any subdifferential for which the norm is smooth, thus providing a new insight on the connection between the smoothness of norms and the subdifferentiability properties of functions. The proof relies on an adaptation of the ''smooth'' variational principle of Borwein-Preiss. Along the same vein, we derive subdifferential criteria for coercivity, Lipschitz behavior, cone-monotonicity, quasiconvexity, and convexity of lower semicontinuous functions which clarify, unify and extend many existing results for specific subdifferentials.</description>
<date>1995</date>
</dc>
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