Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T18:51:00.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Analyse de récession et résultats de stabilité d'uneconvergence variationnelle, application à la théorie de la dualité en programmationmathématique

Published online by Cambridge University Press:  15 September 2003

Driss Mentagui
Driss Mentagui*
Affiliation:
Laboratoire d'Analyse Convexe et Variationnelle, Systèmes Dynamiques et Processus Stochastiques, Université Ibn Tofail, Faculté des Sciences, Département de Mathématiques, BP. 133, Kenitra, Maroc; [email protected].
Get access

Abstract

Let X be a Banach space and X' its continuous dual. C(X) (resp. C(X')) denotes the set of nonempty convex closed subsets of X (resp. ω*-closed subsets of X') endowed with the topology of uniform convergence of distance functions on bounded sets. This topology reduces to the Hausdorff metric topology on the closed and bounded convex sets [16] and in general has a Hausdorff-like presentation [11]. Moreover, this topology is well suited for estimations and constructive approximations [6-9]. We prove here, that under natural qualification conditions, the stability of the convergence associated to the topology defined on C(X) (resp. C(X')) is preserved by a class of linear transformations. Building on these results, and by identifing each convex function with its epigraph, the stability at the functional level is acquired towards some operations of convex analysis which play a basic role in convex optimization and duality theory. The key hypothesis in the qualification conditions ensuring the functional stability is the notion of inf-local compactness of a convex function introduced in [28] and expressed in the space X' by the quasi-continuity of its conjugate. Then we generalize the stability results of McLinden and Bergstrom [31] and the ones of Beer and Lucchetti [17] in infinite dimension case. Finally we give some applications in convex optimization and mathematical programming in general Banach spaces.

Keywords

Fonction convexeopérateur linéaireconvergence au sens d'Attouch–WetsMosco/épi-convergenceconvergence uniforme sur les bornésinf-(locale) compacitéquasi-continuitécône (fonction) horizondualitéstabilitéapproximation et optimisation.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 297 - 315
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

H. Attouch, Variational convergence for functions and operators. Pitman, London, Appl. Math. Ser. (1984).
H. Attouch et D. Aze, Regularization and approximation of sets and functions in Hilbert spaces, dans Séminaire d'Analyse Numérique, Paper XI. Université Paul Sabatier de Toulouse (1987-1988).
Attouch, H., Aze, D. et Wets, R.J.-B., Convergence of convex-concave saddle functions : Continuity properties of the Legendre-Fenchel transform and applications to convex programming. Ann. Inst. H. Poincaré Anal. Non linéaire 5 (1988) 537-572.
Attouch, H. et Beer, G., On the convergence of subdifferentials of convex functions. Arch. Math. 60 (1993) 389-400. CrossRef
H. Attouch et H. Brezis, Duality for the sum of convex functions in general Banach spaces, Publications AVAMAC. Université de Perpignan, Nos. 84-10. Av. (1984).
Attouch, H. et Wets, R.J.-B., Quantitative stability of variational systems: I. The epigraphical distance. Trans. Amer. Math. Soc. 328 (1991) 695-729.
H. Attouch et R.J.-B. Wets, Quantitative stability of variational systems: II. A framework for nonlinear conditionning, IIASA working paper 88-9. Laxemburg, Austria (1988).
H. Attouch et R.J.-B. Wets, Quantitative stability of variational systems: III. Stability of minimizers, Working paper IIASA. Laxemburg, Austria (1988).
H. Attouch et R.J.-B. Wets, A quantitative approach via epigraphic distance to stability of strong local minimizers, Publications AVAMAC. Université de Perpignan (1987).
D. Aze, Convergences variationnelles et dualité. Applications en calcul des variations et en programmation mathématique, Thèse de Doctorat d'État. Université de Perpignan (1986).
Aze, D. et Penot, J.-P., Operations on convergent families of sets and functions. Optim. 21 (1990) 521-534. CrossRef
B. Bank, J. Guddat, D. Klatte, B. Kummer et K. Tammer, Nonlinear parametric optimization. Akademie Verlag (1982).
Beer, G., Mosco, On convergence of convex sets. Bull. Austral. Math. Soc. 38 (1988) 239-253. CrossRef
Beer, G., Conjugate convex functions and the epi-distance topology. Proc. Amer. Math. Soc. 108 (1990) 117-126. CrossRef
Beer, G., The slice topology: A viable alternative to Mosco convergence in nonreflexive spaces. Nonlinear. Anal. Theo. Meth. Appl. 19 (1992) 271-290. CrossRef
Beer, G. et Lucchetti, R., Convex optimization and the epi-distance topology. Trans. Amer. Math. Soc. 327 (1991) 795-813. CrossRef
Beer, G. et Lucchetti, R., The epi-distance topology: Continuity and stability results with applications to convex optimization problems. Math. Oper. Res. 17 (1992) 715-726. CrossRef
Beer, G. et Thera, M., Attouch-Wets convergence and a differential operator for convex functions. Proc. Amer. Math. Soc. 122 (1994) 851-858. CrossRef
N. Bourbaki, Espaces vectoriels topologiques, Chaps. 1-2. Hermann, Paris (1966).
Burkholder, D.L. et Wijsman, R.A., Optimum properties and admissibility of sequentiel tests. Ann. Math. Statist. 34 (1963) 1-17. CrossRef
C. Castaing et M. Valadier, Convex analysis and measurable multifunctions. Springer, Lecture Notes in Math. 580 (1977).
Dieudonne, J., Sur la séparation des ensembles convexes. Math. Annal. 163 (1966) 1-3. CrossRef
A.L. Dontchev et T. Zolezzi, Well-posed optimization problems. Springer-Verlag, Berlin, Lecture Notes in Math. 1543 (1993).
I. Ekeland et R. Temam, Analyse convexe et problèmes variationnels. Dunod, Paris (1974).
K. El Hajioui, Convergences variationnelles : approximations inf-convolutives généralisées, stabilité et optimisation dans les espaces non réflexifs, Thèse Nationale. Kénitra (2002).
K. El Hajioui et D. Mentagui, Slice convergence : stabilité et optimisation dans les espaces non réflexifs. Preprint.
J. Garsoux, Espaces vectoriels topologiques et distributions. Dunod, Paris (1963).
J.L. Joly, Une famille de topologies et de convergences sur l'ensemble des fonctionnelles convexes, Thèse d'État. Grenoble (1970).
K. Kuratowski, Topology, Vol. I. Academic Press, New York (1966).
P.J. Laurent, Approximation et optimisation. Hermann (1972).
McLinden, L. et Bergstrom, R., Preservation of convergence of convex sets and functions in finite dimensions. Trans. Amer. Math. Soc. 268 (1981) 127-142. CrossRef
Mentagui, D., Stability results of a class of well-posed optimization problems. Optim. 36 (1996) 119-138. CrossRef
Mentagui, D., Stabilité de l'épi-convergence en dimension finie. Pub. Inst. Math. 59 (1996) 161-168.
D. Mentagui et K. El Hajioui, Convergences des fonctions convexes et approximations inf-convolutives généralisées. Pub. Inst. Math. (à paraître).
Moreau, J.J., Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. France 93 (1965) 273-299. CrossRef
Mosco, U., Approximation of the solutions of some variational inequalities. Ann. Scuola Normale Sup. Pisa 21 (1967) 373-394.
Mosco, U., Convergence of convex sets and of solutions of variational inequalities. Adv. in Math. 3 (1969) 510-585. CrossRef
Mosco, U., On the continuity of the Young-Fenchel transform. J. Math. Anal. Appl. 35 (1971) 518-535. CrossRef
Robert, R., Convergences de fonctionnelles convexes. J. Math. Anal. Appl. 45 (1974) 533-555. CrossRef
R.T. Rockafellar, Convex Analysis. Princeton University Press (1970).
Rockafellar, R.T., Level sets and continuity of conjugate convex functions. Trans. Amer. Math. Soc. 123 (1966) 46-63. CrossRef
R.T. Rockafellar et R.J.-B. Wets, Variational analysis. Springer (1998).
Salinetti, G. et Wets, R.J.-B., On the relations between two types of convergence for convex functions. J. Math. Anal. Appl. 60 (1977) 211-226. CrossRef
Y. Sonntag, Convergence au sens de Mosco : théorie et applications à l'approximation des solutions d'inéquations, Thèse d'État. Université de Provence, Marseille (1982).
Sonntag, Y. et Zalinescu, C., Set convergences: An attempt of classification. Trans. Amer. Math. Soc. 340 (1993) 199-226. CrossRef
B. Van Cutsem, Problems of convergence in stochastic linear programming, dans Techniques of optimization, édité parBalakrishnan. Academic Press, New York (1972) 445-454.
R.J.-B. Wets, A formula for the level sets of epi-limits and some applications. Mathematical theories of optimization, édité par J.P. Cecconi et T. Zolezzi. Springer, Lecture Notes in Math. 983 (1983).
Wijsman, R.A., Convergence of sequences of convex sets, cones and functions. Bull. Amer. Math. Soc. 70 (1964) 186-188. CrossRef
Wijsman, R.A., Convergence of sequences of convex sets, cones and functions II. Trans. Amer. Math. Soc. 123 (1966) 32-45. CrossRef