Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-07-06T23:42:18.486Z Has data issue: false hasContentIssue false
  • English
  • Français

A formula on the subdifferential of the deconvolution of convex functions

Published online by Cambridge University Press:  17 April 2009

M. Volle
M. Volle
Affiliation:
University of Avignon, 33 rue L Pasteur, 84000 Avignon, France
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is known that, under suitable assumptions, the subdifferential ∂(f □ g) of the infimal convolution of two convex functions f and g coincides with the parallel sum of ∂ f and ∂ g. We prove that a similar formula holds for the subdifferential of the deconvolution of two convex functions: under appropriate hypothesis it coincides with the parallel star-difference of the sub-differentials of the functions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 47 , Issue 2 , April 1993 , pp. 333 - 340
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Attouch, H. and Wets, R.J.B., ‘Epigraphical analysis’, in Analyse non linaire, (Attouch, H., Aubin, J.-P., Clarke, F.H., Ekeland, I., Editors) (Gauthier-Villars, Paris, 1989), pp. 73–100.Google Scholar
[2]Aubin, J.-P. and Ekeland, I., Applied nonlinear analysis (Wiley, New York, 1984).Google Scholar
[3]Ellaia, R., Contribution à l'analyse et l'optimisation de difference de fonctions convexes, Thèse (Université Paul Sabatier, Toulouse, 1984).Google Scholar
[4]Hiriart-Urruty, J.-B., ‘A general formula on the conjugate of the difference of functions’, Canad. Math. Bull. 29 (1986), 482–485.Google Scholar
[5]Hiriart-Urruty, J.-B. and Mazure, M.-L., ‘Formulations variationnelles de l'addition parallèle et de la soustraction parallèle d'opérateurs semi-définis positifs’, C.R. Acad. Sci. Paris Série. I 302 (1986), 527–530.Google Scholar
[6]Laurent, P.-J., Approximation et optimisation (Hermann, Paris, 1972).Google Scholar
[7]Martinez-Legaz, J.E. and Seeger, A.A formula on the approximate subdifferential of the difference of convex functions, Bull. Austral. Math. Soc. 45 (1992), 37–41.Google Scholar
[8]Mazure, M.-L., ‘La soustraction parallèle d'operateurs interpretée comme déconvolution de formes quadratiques convexes’, Optimization 18 (1987), 465–484.Google Scholar
[9]Mazure, M.-L. and Voile, M., ‘Equations inf-convolutives et conjugaison de Moreau-Fenchel’, Ann. Fac. Sci. Toulouse Math 12 (1991), 103–126.Google Scholar
[10]Moreau, J.-J., ‘Inf-convolution, sous-additivité, convexité des fonctions numériques’, J. Math. Pures Appl. 49 (1970), 109–154.Google Scholar
[11]Moudafi, A., Convolution des opérateurs monotones et convergence variationnelle (Université de Clermont-Ferrand II, 1991). (Preprint).Google Scholar
[12]Passty, G.B., ‘The parallel sum of nonlinear monotone operators’, Nonlinear Anal. 10 (1986), 215–227.Google Scholar
[13]Pekarev, E.L. and Smul'jan, L., ‘Parallel addition and parallel subtraction of operators’, Math. USSR-Izv. 10 (1976), 351–370.CrossRefGoogle Scholar
[14]Penot, J.-P., ‘Calcul sous-différentiel et optimisation’, J. Funct. Anal. 27 (1978), 248–276.Google Scholar
[15]Volle, M., ‘Concave duality: application to problems dealing with difference of functions’, Math. Programming 41 (1988), 261–278.Google Scholar