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Generalised second-order derivatives of convex functions in reflexive Banach spaces

Published online by Cambridge University Press:  17 April 2009

James Louis Ndoutoume  and
Michel Théra
James Louis Ndoutoume
Affiliation:
Institut Africain d'InformatiqueB.P. 2263Libreville Gabon and LACO URA-CNRS 1586 Université de Limoges F-87060 Limoges, Cedex, France
Michel Théra
Affiliation:
LACO URA-CNRS 1586Université de LimogesF-87060 Limoges, CedexFrance e-mail: [email protected]
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Abstract

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Generalised second-order derivatives introduced by Rockafellar in the finite dimensional setting are extended to convex functions defined on reflexive Banach spaces. Our approach is based on the characterisation of convex generalised quadratic forms defined in reflexive Banach spaces, from the graph of the associated subdifferentials. The main result which is obtained is the exhibition of a particular generalised Hessian when the function admits a generalised second derivative. Some properties of the generalised second derivative are pointed out along with further justifications of the concept.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 51 , Issue 1 , February 1995 , pp. 55 - 72
Copyright
Copyright © Australian Mathematical Society 1995

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