Advertisement

Generalized Monotonicity of Subdifferentials and Generalized Convexity

  • J. P. Penot
  • P. H. Sach
Article

Abstract

Characterizations of convexity and quasiconvexity of lower semicontinuous functions on a Banach space X are presented in terms of the contingent and Fréchet subdifferentials. They rely on a general mean-value theorem for such subdifferentials, which is valid in a class of spaces which contains the class of Asplund spaces.

Convexity generalized convexity monotone functions pseudoconvexity pseudomonotone functions quasiconvexity quasimonotone functions subdifferentials 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    AVRIEL, M., DIEWERT, W. E., SCHAIBLE, S., and ZHANG, I., Generalized Concavity, Plenum Press, New York, New York, 1988.Google Scholar
  2. 2.
    CAMBINI, A., CASTAGNOLI, E., MARTEIN, L., MAZZOLENI, P., and SCHAIBLE, S., Editors, Generalized Convexity and Fractional Programming with Economic Applications, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 305, 1990.Google Scholar
  3. 3.
    CROUZEIX, J. P., Contributions à l'Etude des Fonctions Quasiconvexes, Thèse d'Etat, Université de Clermont-Ferrand III, Clermont-Ferrand, France.Google Scholar
  4. 4.
    SCHAIBLE, S., and ZIEMBA, W. T., Editors, Generalized Concavity in Optimization and Economics, Academic Press, New York, New York, 1981.Google Scholar
  5. 5.
    KARAMARDIAN, S., Complementarity over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–454, 1976.Google Scholar
  6. 6.
    KARAMARDIAN, S., and SCHAIBLE, S., Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37–46, 1990.Google Scholar
  7. 7.
    KOMLOSI, S., Some Properties of Nondifferentiable Pseudoconvex Functions, Mathematical Programming, Vol. 26, pp. 232–237, 1983.Google Scholar
  8. 8.
    KOMLOSI, S., Monotonicity and Quasimonotonicity for Multifunctions, Optimization of Generalized Convex Problems in Economics, Edited by P. Mazzoleni, University of Pisa, Pisa, Italy, pp. 27–38, 1994.Google Scholar
  9. 9.
    KOMLOSI, S., On Generalized Upper Quasidifferentiability, Nonsmooth Optimization: Methods and Applications, Edited by F. Giannessi, Gordon and Breach, London, England, pp. 189–200, 1992.Google Scholar
  10. 10.
    MANGASARIAN, O. L., Pseudoconvex Functions, SIAM Journal on Control, Vol. 3, pp. 281–290, 1965.Google Scholar
  11. 11.
    MANGASARIAN, O. L., Nonlinear Programming, McGraw-Hill, New York, New York, 1969.Google Scholar
  12. 12.
    AUSSEL D., Théorème de la Valeur Moyenne et Convexité Généralisée en Analyse Non-Régulière, Thèse, Université de Clermont-Ferrand, Clermont-Ferrand, France, 1994.Google Scholar
  13. 13.
    AUSSEL, D., CORVELLEC, J. N., and LASSONDE, M., Subdifferential Characterization of Quasiconvexity and Convexity, Journal of Convex Analysis, Vol. 1, pp. 195–201, 1994.Google Scholar
  14. 14.
    CORREA, R., JOFRE, A., and THIBAULT, L., Characterization of Lower Semicontinuous Convex Functions, Proceedings of the American Mathematical Society, Vol. 116, pp. 67–72, 1992.Google Scholar
  15. 15.
    CORREA, R., JOFRE, A., and THIBAULT, L., Subdifferential Monotonicity as Characterization of Convex Functions, Numerical Functional Analysis and Optimization, Vol. 15, pp. 531–535, 1994.Google Scholar
  16. 16.
    DIEWERT, W. E., Alternative Characterizations of Six Kinds of Quasiconcavity in the Nonfifferentiable Case with Applications to Nonsmooth Programming, Generalized Concavity in Optimization and Economics, Edited by S. Schaible and W. T. Ziemba, Academic Press, New York, New York, 1981.Google Scholar
  17. 17.
    ELLAIA, R., and HASSOUNI, A., Characterization of Nonsmooth Functions through Their Generalized Gradients, Optimization, Vol. 22, pp. 401–416, 1991.Google Scholar
  18. 18.
    GIORGI, G., and KOMLOSI, S., Dini Derivatives in Optimization, Parts 1 and 2, Rivista di Matematica per le Scienze Economiche e Sociali, Vol. 15,No. 1, pp. 3–30, 1993 and Vol. 15, No. 2, pp. 3–24, 1993.Google Scholar
  19. 19.
    HASSOUNI, A., Sous-Différentiels des Fonctions Quasiconvexes, Thèse de Troisième Cycle, Université P. Sabatier, Toulouse, France, 1983.Google Scholar
  20. 20.
    HASSOUNI, A., Quasimonotone Multifunctions: Applications to Optimality Conditions in Quasiconvex Programming, Numerical Functional Analysis and Optimization, Vol. 13, pp. 267–275, 1992.Google Scholar
  21. 21.
    KARAMARDIAN, S., SCHAIBLE, S., and CROUZEIX, J. P., Characterizations of Generalized Monotone Maps, Journal of Optimization Theory and Applications, Vol. 76, pp. 399–413, 1993.Google Scholar
  22. 22.
    KOMLOSI, S., Generalized Monotonicity and Generalized Convexity, Journal of Optimization Theory and Applications, Vol. 84, pp. 361–376, 1995.Google Scholar
  23. 23.
    KOMLOSI, S., Monotonicity and Quasimonotonicity in Nonsmooth Analysis, Working Paper, University of Pisa, Pisa, Italy, 1994.Google Scholar
  24. 24.
    LUC, D. T., Characterizations of Quasiconvex Functions, Bulletin of the Australian Mathematical Society, Vol. 48, pp. 393–405, 1993.Google Scholar
  25. 25.
    LUC, D. T., and SWAMINATHAN, S., A Characterization of Convex Functions, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 30, pp. 697–701, 1993.Google Scholar
  26. 26.
    PENOT, J. P., and QUANG, P. H., On Generalized Convex Functions and Generalized Monotonicity of Set-Valued Maps, Journal of Optimization Theory and Applications (to appear).Google Scholar
  27. 27.
    PENOT, J. P., Generalized Convexity in the Light of Nonsmooth Analysis, Proceedings of the 7th French-German Conference on Optimization, Dijon, France, 1995; Edited by R. Duriez and C. Michelot, Lecture Notes in Mathematical Systems and Economics, Springer Verlag, Berlin, Germany, Vol. 421, pp. 269–290, 1995.Google Scholar
  28. 28.
    POLIQUIN, R., Subgradient Monotonicity and Convex Functions, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 14, pp. 305–317, 1990.Google Scholar
  29. 29.
    SCHAIBLE, S., Generalized Monotone Maps, Nonsmooth Optimization: Methods and Applications, Edited by F. Giannessi, Gordon and Breach Science Publishers, Amsterdam, Holland, pp. 392–408, 1992.Google Scholar
  30. 30.
    CRANDALL, M. G., ISHII, H., and LIONS, P. L., User's Guide to Viscosity Solutions of Second-Order Partial Differential Equations, Bulletin of the American Mathematical Society, Vol. 27, pp. 1–67, 1992.Google Scholar
  31. 31.
    DEGIOVANNI, M., MARINO, A., and TOSQUES, M., Evolution Equations with Lack of Convexity, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 9, pp. 1401–1443, 1985.Google Scholar
  32. 32.
    IOFFE, A. D., Subdifferentiability Spaces and Nonsmooth Analysis, Bulletin of the American Mathematical Society, Vol. 10, pp. 87–90, 1984.Google Scholar
  33. 33.
    PENOT, J. P., Mean-Value Theorem with Small Subdifferentials, Journal of Optimization Theory and Applications, Vol. 94, pp. 209–221, 1997.Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • J. P. Penot
    • 1
  • P. H. Sach
    • 2
  1. 1.Laboratory of Applied MathematicsUniversity of PauPauFrance
  2. 2.Hanoi Institute of MathematicsBo Ho, HanoiVietnam

Personalised recommendations