Analyse de récession et résultats de stabilité d'une convergence variationnelle, application à la théorie de la dualité en programmation mathématique

Driss Mentagui

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 9, page 297-315
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Mentagui, Driss. " Analyse de récession et résultats de stabilité d'une convergence variationnelle, application à la théorie de la dualité en programmation mathématique ." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 297-315. <http://eudml.org/doc/90697>.

@article{Mentagui2010,
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. },
author = {Mentagui, Driss},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Fonction convexe; opérateur linéaire; convergence au sens d'Attouch–Wets; Mosco/épi-convergence; convergence uniforme sur les bornés; inf-(locale) compacité; quasi-continuité; cône (fonction) horizon; dualité; stabilité; approximation et optimisation. ; convex function; linear operator; Attouch-Wets convergence; Moso convergence; epiconvergence; uniform convergence on bounded sets; inf-(local) compacity; quasi-continuity; horizon (function) cone; duality; stability; approximation and optimization.},
language = {fre},
month = {3},
pages = {297-315},
publisher = {EDP Sciences},
title = { Analyse de récession et résultats de stabilité d'une convergence variationnelle, application à la théorie de la dualité en programmation mathématique },
url = {http://eudml.org/doc/90697},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Mentagui, Driss
TI - Analyse de récession et résultats de stabilité d'une convergence variationnelle, application à la théorie de la dualité en programmation mathématique
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 297
EP - 315
AB - 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.
LA - fre
KW - Fonction convexe; opérateur linéaire; convergence au sens d'Attouch–Wets; Mosco/épi-convergence; convergence uniforme sur les bornés; inf-(locale) compacité; quasi-continuité; cône (fonction) horizon; dualité; stabilité; approximation et optimisation. ; convex function; linear operator; Attouch-Wets convergence; Moso convergence; epiconvergence; uniform convergence on bounded sets; inf-(local) compacity; quasi-continuity; horizon (function) cone; duality; stability; approximation and optimization.
UR - http://eudml.org/doc/90697
ER -

References

top
  1. H. Attouch, Variational convergence for functions and operators. Pitman, London, Appl. Math. Ser. (1984).  
  2. 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).  
  3. H. Attouch, D. Aze et R.J.-B. Wets, Convergence of convex-concave saddle functions : Continuity properties of the Legendre-Fenchel transform and applications to convex programming. Ann. Inst. H. Poincaré Anal. Non linéaire5 (1988) 537-572.  
  4. H. Attouch et G. Beer, On the convergence of subdifferentials of convex functions. Arch. Math.60 (1993) 389-400.  
  5. 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).  
  6. H. Attouch et R.J.-B. Wets, Quantitative stability of variational systems: I. The epigraphical distance. Trans. Amer. Math. Soc.328 (1991) 695-729.  
  7. 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).  
  8. H. Attouch et R.J.-B. Wets, Quantitative stability of variational systems: III. Stability of minimizers, Working paper IIASA. Laxemburg, Austria (1988).  
  9. 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).  
  10. D. Aze, Convergences variationnelles et dualité. Applications en calcul des variations et en programmation mathématique, Thèse de Doctorat d'État. Université de Perpignan (1986).  
  11. D. Aze et J.-P. Penot, Operations on convergent families of sets and functions. Optim.21 (1990) 521-534.  
  12. B. Bank, J. Guddat, D. Klatte, B. Kummer et K. Tammer, Nonlinear parametric optimization. Akademie Verlag (1982).  
  13. G. Beer, On Mosco convergence of convex sets. Bull. Austral. Math. Soc.38 (1988) 239-253.  
  14. G. Beer, Conjugate convex functions and the epi-distance topology. Proc. Amer. Math. Soc.108 (1990) 117-126.  
  15. G. Beer, The slice topology: A viable alternative to Mosco convergence in nonreflexive spaces. Nonlinear. Anal. Theo. Meth. Appl.19 (1992) 271-290.  
  16. G. Beer et R. Lucchetti, Convex optimization and the epi-distance topology. Trans. Amer. Math. Soc.327 (1991) 795-813.  
  17. G. Beer et R. Lucchetti, The epi-distance topology: Continuity and stability results with applications to convex optimization problems. Math. Oper. Res.17 (1992) 715-726.  
  18. G. Beer et M. Thera, Attouch-Wets convergence and a differential operator for convex functions. Proc. Amer. Math. Soc.122 (1994) 851-858.  
  19. N. Bourbaki, Espaces vectoriels topologiques, Chaps. 1-2. Hermann, Paris (1966).  
  20. D.L. Burkholder et R.A. Wijsman, Optimum properties and admissibility of sequentiel tests. Ann. Math. Statist.34 (1963) 1-17.  
  21. C. Castaing et M. Valadier, Convex analysis and measurable multifunctions. Springer, Lecture Notes in Math. 580 (1977).  
  22. J. Dieudonne, Sur la séparation des ensembles convexes. Math. Annal.163 (1966) 1-3.  
  23. A.L. Dontchev et T. Zolezzi, Well-posed optimization problems. Springer-Verlag, Berlin, Lecture Notes in Math. 1543 (1993).  
  24. I. Ekeland et R. Temam, Analyse convexe et problèmes variationnels. Dunod, Paris (1974).  
  25. K. El Hajioui, Convergences variationnelles : approximations inf-convolutives généralisées, stabilité et optimisation dans les espaces non réflexifs, Thèse Nationale. Kénitra (2002).  
  26. K. El Hajioui et D. Mentagui, Slice convergence : stabilité et optimisation dans les espaces non réflexifs. Preprint.  
  27. J. Garsoux, Espaces vectoriels topologiques et distributions. Dunod, Paris (1963).  
  28. J.L. Joly, Une famille de topologies et de convergences sur l'ensemble des fonctionnelles convexes, Thèse d'État. Grenoble (1970).  
  29. K. Kuratowski, Topology, Vol. I. Academic Press, New York (1966).  
  30. P.J. Laurent, Approximation et optimisation. Hermann (1972).  
  31. L. McLinden et R. Bergstrom, Preservation of convergence of convex sets and functions in finite dimensions. Trans. Amer. Math. Soc.268 (1981) 127-142.  
  32. D. Mentagui, Stability results of a class of well-posed optimization problems. Optim.36 (1996) 119-138.  
  33. D. Mentagui, Stabilité de l'épi-convergence en dimension finie. Pub. Inst. Math.59 (1996) 161-168.  
  34. D. Mentagui et K. El Hajioui, Convergences des fonctions convexes et approximations inf-convolutives généralisées. Pub. Inst. Math. (à paraître).  
  35. J.J. Moreau, Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. France93 (1965) 273-299.  
  36. U. Mosco, Approximation of the solutions of some variational inequalities. Ann. Scuola Normale Sup. Pisa21 (1967) 373-394.  
  37. U. Mosco, Convergence of convex sets and of solutions of variational inequalities. Adv. in Math.3 (1969) 510-585.  
  38. U. Mosco, On the continuity of the Young-Fenchel transform. J. Math. Anal. Appl.35 (1971) 518-535.  
  39. R. Robert, Convergences de fonctionnelles convexes. J. Math. Anal. Appl.45 (1974) 533-555.  
  40. R.T. Rockafellar, Convex Analysis. Princeton University Press (1970).  
  41. R.T. Rockafellar, Level sets and continuity of conjugate convex functions. Trans. Amer. Math. Soc.123 (1966) 46-63.  
  42. R.T. Rockafellar et R.J.-B. Wets, Variational analysis. Springer (1998).  
  43. G. Salinetti et R.J.-B. Wets, On the relations between two types of convergence for convex functions. J. Math. Anal. Appl.60 (1977) 211-226.  
  44. 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).  
  45. Y. Sonntag et C. Zalinescu, Set convergences: An attempt of classification. Trans. Amer. Math. Soc.340 (1993) 199-226.  
  46. B. Van Cutsem, Problems of convergence in stochastic linear programming, dans Techniques of optimization, édité parBalakrishnan. Academic Press, New York (1972) 445-454.  
  47. 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).  
  48. R.A. Wijsman, Convergence of sequences of convex sets, cones and functions. Bull. Amer. Math. Soc.70 (1964) 186-188.  
  49. R.A. Wijsman, Convergence of sequences of convex sets, cones and functions II. Trans. Amer. Math. Soc.123 (1966) 32-45.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.