Abstract

Let X be a decomposable set, h a convex function defined on a finite dimensional vector space. We show that under transversality assumptions the problem of minimization on X: inf {f(x)+h(g(x))} admits Lagrange multipliers. We consider the case where f is a scalar integral functional and g is a vector valued integral functional. These properties are related to growth conditions between integrands.

Author information

Contact details are reproduced from the original publication and may be historical.

A. Bourass

Faculté des Sciences, Université Mohamed V, Rue Ibn Batouta, Rabat, Maroc

E. Giner

Laboratoire MIP, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse 04, France

Keywords

  • Performance function
  • multipliers
  • stability
  • convex like functions
  • measurable integrands
  • richness
  • integral functional
  • growth conditions

Mathematics Subject Classification

46E30; 28A20, 60B12

References

  1. V. I. Arkin: An infinite dimensional analog of non convex programming problems, Kibernetika (2) (in Russian) (1967) 87–93; Cybernetics (3) (1967) 70–75.https://doi.org/10.1007/BF01078756Google Scholar
  2. J. P. Aubin: Mathematical Methods of Game and Economic Theory, North Holland, revised edition (1982).Google Scholar
  3. J. P. Aubin, I. Ekeland: Minimisation de critères intégraux, C. R. Acad. Sci., Paris, Sér. A 281 (1975) 285–288.Google Scholar
  4. M. S. Bazaraa, C. M. Shetty: Non Linear Programming Theory and Algorithms, John Wiley, New York (1979).Google Scholar
  5. H. Berliocchi, J. M. Lasry: Integrandes normales et mesures paramétrées en calcul des variations, Bull. Soc. Math. France 101 (1973) 129–184.https://doi.org/10.24033/bsmf.1755Google Scholar
  6. A. Bourass: Comparaison de fonctionnelles intégrales. Conditions de croissance caractéristiques et application à la semi-continuité dans les espaces intégraux de type Orlicz, Thèse d’Etat, Perpignan (1983).Google Scholar
  7. C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Lecture Notes Math. 580, Springer-Verlag, Berlin et al. (1977).https://doi.org/10.1007/BFb0087685Google Scholar
  8. J. Diestel, J. J. Uhl: Vector Measures, Mathematical Surveys 15, American Mathematical Society (1977).https://doi.org/10.1090/surv/015Google Scholar
  9. A. Fougères: Comparaison de fonctionnelles intégrales sur les sélections d’une multiapplication mesurable: théorème d’approximation et condition de croissance liée à l’inclusion des sections, Séminaire d’Analyse Convexe, Montpellier, no. 9 (1982).Google Scholar
  10. A. Fougères, R. Vaudene: Comparaison de fonctionnelles intégrales; application aux opérateurs intégrands entre espaces d’Orlicz, Séminaire d’Analyse Convexe, Montpellier, no. 3 (1977).Google Scholar
  11. E. Giner: Local minimizers of integral functionals are global minimizers, Proceedings of the American Mathematical Society 123 (1995) 755–757.https://doi.org/10.1090/S0002-9939-1995-1254839-1Google Scholar
  12. E. Giner: Minima sous contrainte de fonctionnelles intégrales, C. R. Acad. Sci., Paris Sér. I 321 (1995) 429–431.Google Scholar
  13. E. Giner: On Pareto minima of vector-valued integral functionals, Optimization 48 (2000) 107–116.https://doi.org/10.1080/02331930008844496Google Scholar
  14. M. R. Hestenes: Optimization Theory. The Finite Dimentional Case, Robert Krieger Publishing Company, New York (1981).Google Scholar
  15. F. Hiai: Representation of additive functionals on vector valued normed Köthe space, Kodai Math. J. 2 (1979) 300–313.https://doi.org/10.2996/kmj/1138036062Google Scholar
  16. J. B. Hiriart-Urruty: ε-subdifferential calculus, in: J. P. Aubin and R. Vinter (eds.), Pitman (1982) 43–92.Google Scholar
  17. R. B. Holmes: Geometric Functional Analysis and its Applications, Springer-Verlag, Berlin et al. (1975).https://doi.org/10.1007/978-1-4684-9369-6Google Scholar
  18. J. J. Moreau: Fonctionnelles Convexes, Lecture Notes, Séminaire "Equations aux dérivées partielles", Collège de France, Paris (1966).Google Scholar
  19. J. Neveu: Bases Mathématiques de Calcul des Probabilités, Masson (1964).Google Scholar
  20. J. P. Penot: Multipliers and generalized derivatives of performance functions, J. Optimization Theory Appl. (1997) 609–618.https://doi.org/10.1023/A:1022699214712Google Scholar
  21. R. T. Rockafellar: Convex Analysis, Princeton University Press (1970).https://doi.org/10.1515/9781400873173Google Scholar
  22. R. T. Rockafellar: Integral functionals, normal integrands and measurable selections, Springer-Verlag, Lect. Notes Math. 543 (1976) 157–206.https://doi.org/10.1007/BFb0079944Google Scholar
  23. M. Valadier: Intégration de convexes fermés notamment d’epigraphes, Inf-convolution continue, Rev. Franc. Inform. Rech. Oper. 4 (1970) 57–73.https://doi.org/10.1051/m2an/197004R200571Google Scholar
  24. M. Volle: Sur quelques formules de dualité convexe et non convexe, Set-Valued Analysis 2 (1994) 369–379.https://doi.org/10.1007/BF01027112Google Scholar
  25. M. Moussaoui, M. Volle: Sur la quasi-continuité et les fonctions unies en dualité convexe, C. R. Acad. Sci., Paris, Sér. I 322 (1996) 839–844.Google Scholar
  26. M. Moussaoui, M. Volle: Quasicontinuity and united functions in convex duality theory, Communications on Applied Nonlinear Analysis 4, No. 4 (1997) 73–89.Google Scholar
  27. J. M. Borwein, A. S. Lewis: Partially convex programming, Pt1: Quasi relative interiors and duality theory, Mathematical Programming 57 (1992) 15–48.https://doi.org/10.1007/BF01581072Google Scholar
  28. I. Ekeland, R. Temam: Analyse Convexe et Problèmes Variationnels, Dunod-Gauthier-Villars (1974).Google Scholar
  29. F. Hiai, H. Umegaki: Integrals, conditional expectations, and martingales of multivalued functions, Journal of Multivariate Analysis 7 (1977) 149–182.https://doi.org/10.1016/0047-259X(77)90037-9Google Scholar
  30. D. G. Luenberger: Optimization by Vector Space Methods, John Wiley (1969).Google Scholar

Suggested citation

A. Bourass, E. Giner. “Kuhn-Tucker Conditions and Integral Functionals.” Journal of Convex Analysis 8 (2001), No. 2, 533–553. https://doi.org/10.68381/jca08028

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2001. Rights now held by Banach Press LLC.