Journal of Optimization Theory and Applications

, Volume 67, Issue 1, pp 175–184 | Cite as

A stability result in quasi-convex programming

  • D. Aze
  • M. Volle
Technical Note


Combining a result on the lower semicontinuity of the intersection of two convex-valued multifunctions and the level set approach of epi-convergence, we obtain results on the epi-upper semicontinuity of the supremum and the sum of two families of quasi-convex functions. As a consequence, we give some condition ensuring the stability of a quasi-convex program under a perturbation of the objective functions and the constraint sets.

Key Words

Convexity quasi-convexity set convergence epi-semicontinuity epi-convergence value functions 


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  1. 1.
    Lechicki, A., andSpakowski, A.,A Note on Intersection of Lower Semicontinuous Multifunctions, Proceedings of the American Mathematical Society, Vol. 95, pp. 119–122, 1985.Google Scholar
  2. 2.
    Lucchetti, R., andPatrone, F.,Closure and Upper Semicontinuous Results in Mathematical Programming, Optimization, Vol. 17, pp. 619–628, 1986.Google Scholar
  3. 3.
    Moreau, J. J.,Intersection of Moving Convex Sets in a Normed Space, Mathematica Scandinavica, Vol. 36, pp. 159–173, 1975.Google Scholar
  4. 4.
    Aubin, J. P., Book in preparation.Google Scholar
  5. 5.
    Aze, D., andPenot, J. P.,Operations on Convergent Families of Sets and Functions, Optimization, Vol. 21, pp. 1–14, 1990.Google Scholar
  6. 6.
    Penot, J. P.,Preservation of Persistence and Stability under Intersection and Operations (to appear).Google Scholar
  7. 7.
    Rockafellar, R.-T., andWets, R. J. B.,Variational Systems: An Introduction, Multifunctions and Integrands, Edited by G. Salinetti, Springer-Verlag, Berlin, Germany, 1984.Google Scholar
  8. 8.
    De Giorgi, E., andFranzoni, T., Su un Tipo di Convergenza Variazionale, Atti della Accademia Nazionale dei Lincei, Vol. 8, pp. 842–850, 1975.Google Scholar
  9. 9.
    Dolecki, S.,Tangency and Differentiation: Some Applications of Convergence Theory, Annali di Matematica Pura ed Applicata, Vol. 130, pp. 223–255, 1984.Google Scholar
  10. 10.
    Volle, M., Convergence en Niveaux et en Epigraphes, Notes aux Comptes Rendus de l'Académie des Sciences de Paris, Vol. 299, pp. 295–298, 1984.Google Scholar
  11. 11.
    Wets, R. J. B.,A Formula for the Level Sets of Epilimits and Some Applications, Mathematical Theories of Optimization, Edited by J. Cecconi and T. Zolezzi, Springer-Verlag, Berlin, Germany, 1983.Google Scholar
  12. 12.
    Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.Google Scholar
  13. 13.
    Flachs, J., andPollatschek, M. A.,Duality Theorem for Certain Programs Involving Minimum or Maximum Operations, Mathematical Programming, Vol. 16, pp. 348–370, 1970.Google Scholar
  14. 14.
    Volle, M.,Deux Formules de Dualité Englobant celles de Fenchel et de Toland, Publications Mathématiques de l'Université de Limoges, Limoges, France, 1986.Google Scholar
  15. 15.
    MacLinden, L., andBergstrom, R. C.,Preservation of Convergence of Convex Sets and Functions in Finite Dimensions, Transaction of the American Mathematical Society, Vol. 268, pp. 127–141, 1981.Google Scholar
  16. 16.
    Rockafellar, R. T.,Conjugate Duality and Optimization, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1974.Google Scholar
  17. 17.
    Moreau, J.-J.,Fonctionnelles Convexes, Lecture Notes, Collège de France, Paris, France, 1966.Google Scholar
  18. 18.
    Attouch, H.,Variational Convergence for Functions and Operators, Pitman, London, England, 1984.Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • D. Aze
    • 1
  • M. Volle
    • 2
  1. 1.Department of MathematicsUniversity of PerpignanPerpignanFrance
  2. 2.Department of MathematicsUniversity of AvignonAvignonFrance

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