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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: thera@cix.cict.fr
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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

References

[1]Attouch, H., Variational convergences for functions and operators (Pitman, London, 1984).Google Scholar
[2]Attouch, H., Ndoutoume, J.-L. and Théra, M., ‘Epigraphical convergence of functions and convergence of their derivatives in Banach spaces’, Séminaire d'Analyse Convexe de Montpellier, Exposé No.9, 9.1–9.45, 1990.Google Scholar
[3]Attouch, H. and Théra, M., ‘Convergences en analyse multivoque et unilatérall’, MATAPLI 36 (1993), 35–52.Google Scholar
[4]Barbu, V., Optimal control of variational inequalities (Pitman Advanced Publishing Program, 1984).Google Scholar
[5]Borwein, J.M. and Noll, D., ‘Second order differentiability of convex functions in Hilbert spaces’, Trans. Amer. Math. Soc. 342 ((1994)), 43–82.Google Scholar
[6]Day, M.M., Normed linear spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1963).Google Scholar
[7]Do, V.N., ‘Generalized second-order derivatives of convex functions in reflexive Banach spaces’, Trans. Amer. Math. Soc. 134 (1992), 281–301.Google Scholar
[8]Jameson, G.J.O., Topology and normed spaces (Chapman and Hall, London, 1974).Google Scholar
[9]Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces, Lecture Notes in Mathematics 338 (Springer-Verlag, Berlin, Heidelberg, New York, 1973).Google Scholar
[10]Navry, S., ‘Formes quadratiques généralisées’, C.R. Acad. Sci. Paris, série A 271 (1970), 1054–1057.Google Scholar
[11]Ndoutoume, J.L., Epi-convergence en calcul différentiel généralisé Résultats de convergence et Approximation, Thése de doctorat (Université de Perpignan, 1987).Google Scholar
[12]Ndoutoume, J.L., ‘Optimality conditions for control problems governed by variational inequalities’, Math. Oper. Res. 19, 676–690.Google Scholar
[13]Rockafellar, R.T., ‘Maximal monotone relations and the second order derivative of non-smooth functions’, Ann. Inst. H. Poincaré, Anal. Non Linéaire 2 (1985), 167–184.CrossRefGoogle Scholar
[14]Rockafellar, R.T., ‘First and second order epi-differentiability in nonlinear programming’, Trans. Amer. Math. Soc. 307 (1988), 75–108.Google Scholar
[15]Rockafellar, R.T., ‘Generalized second derivatives of convex functions and saddle functions’, Trans. Amer. Math. Soc. 320 (1990), 810–822.Google Scholar
[16]Rockafellar, R.T. and Wets, R.J.-B., Variational Analysis (to appear).Google Scholar
[17]Rockafellar, R.T., ‘Characterization of the subdifferentials of convex functions’, Pacific J. Math 17 (1966), 497–510.CrossRefGoogle Scholar
[18]Seeger, A., ‘Limiting behaviour of the approximate second-order subdifferential of a convex function’, J. Optim. Theory Appl. 74 (1992), 527–544.Google Scholar