Skip to main content
SpringerLink
Log in

Preservation of persistence and stability under intersections and operations, part 1: Persistence

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

New results about convergence of sets and functions in possibly infinite-dimensional spaces are derived in a simple and unified way from two results dealing with the continuity with respect to a parameter of the intersection of two convex sets depending on this parameter.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Contesse, L., andPenot, J. P.,Continuity of the Fenchel Correspondence and of the Approximate Subdifferential by Geometrical Methods, Journal of Mathematical Analysis and Applications, Vol. 156, pp. 305–328, 1991.

    Google Scholar 

  2. Dolecki, S.,Metrically Upper-Semicontinuous Multifunctions and Their Intersections, Technical Summary Report 2035, University of Wisconsin, Madison, 1980.

    Google Scholar 

  3. Lucchetti, R., andPatrone, F.,Closure and Upper Semicontinuity Results in Mathematical Programming, Nash and Economic Equilibria, Optimization, Vol. 17, pp. 619–628, 1985.

    Google Scholar 

  4. McLinden, L., andBergstrom, R. C.,Preservation of Convergence of Convex Sets and Functions in Finite Dimensions, Transactions of the American Mathematical Society, Vol. 268, pp. 127–141, 1981.

    Google Scholar 

  5. Momal, P.,Théorèmes de Maximum, Comptes Rendus de l'Académie des Sciences de Paris, Vol. 278A, pp. 905–907, 1974.

    Google Scholar 

  6. Moreau, J.-J.,Intersection of Moving Convex Sets in a Normed Space, Mathematica Scandinavica, Vol. 36, pp. 159–173, 1975.

    Google Scholar 

  7. Mosco, U.,Convergence of Convex Sets and of Solutions of Variational Inequalities, Advances in Mathematics, Vol. 3, pp. 540–585, 1960.

    Google Scholar 

  8. Penot, J. P.,Convergence of Feasible Sets, Unpublished Manuscript, University of Pau, 1980.

  9. Penot, J. P.,Transversality and Convexity (manuscript in preparation).

  10. Penot, J. P., andSterna-Karwat, A.,Parametrized Multicriteria Optimization: Continuity and Closedness of Optimal Multifunctions, Journal of Mathematical Analysis and Applications, Vol. 120, pp. 150–168, 1986.

    Google Scholar 

  11. Robert, R.,Convergence de Fonctionnelles Convexes, Comptes Rendus de l'Académie des Sciences de Paris, Vol. 276A, pp. 727–729, 1973.

    Google Scholar 

  12. Rolewicz, S.,On Intersections of Multifunctions, Mathematische Operationsforschung und Statistics, Series Optimization, Vol. 11, pp. 3–11, 1980.

    Google Scholar 

  13. Attouch, H.,Variational Convergence for Functions and Operators, Pitman, London, England, 1984.

    Google Scholar 

  14. Attouch, H., Aze, D., andWets, R. J. B.,Convergence of Convex-Concave Saddle Functions: Continuity Properties of the Legendre - Fenchel Transform with Applications to Convex Programming and Mechanics, Annales de l'Institut H. Poincaré, Analyse Non Linéaire, Vol. 5, pp. 537–572, 1988.

    Google Scholar 

  15. Attouch, H., andBrezis, H.,Duality of the Sum of Convex Functions in General Banach Spaces, Aspects of Mathematics and Its Applications, Edited by J. Barroso, North-Holland, Amsterdam, Netherlands, pp. 125–133, 1986.

    Google Scholar 

  16. Attouch, H., andWets, R. J. B.,Isometries for the Legendre-Fenchel Transform, Transactions of the American Mathematical Society, Vol. 296, pp. 33–60, 1986.

    Google Scholar 

  17. Rockafellar, R. T., andWets, R. J.-B.,Variational System: An Introduction, Multifunctions and Integrands, Edited by G. Salinetti, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Vol. 1091, pp. 1–53, 1984.

    Google Scholar 

  18. Volle, M.,Contributions à la Dualité en Optimisation et à l'Epiconvergence, Thèse d'Etat, University of Pau, 1986.

  19. Zolezzi, T.,On Stability Analysis in Mathematical Programming, Mathematical Programming Study, Vol. 21, pp. 227–242, 1984.

    Google Scholar 

  20. Robinson, S.,Local Epicontinuity and Local Optimization, Mathematical Programming, Vol. 37, pp. 208–222, 1987.

    Google Scholar 

  21. Dolecki, S.,Tangency and Differentiation: Some Applications of Convergence Theory, Annali di Matematica Pura ed Applicada, Vol. 130, pp. 223–255, 1982.

    Google Scholar 

  22. 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 

  23. Lechicki, A., andZieminska, J.,On Limits in Spaces of Sets, Bollettino della Unione Matematica Italiana, Vol. 6B, pp. 17–37, 1986.

    Google Scholar 

  24. Aze, D., andVolle, M.,A Stability Result in Quasi-Convex Programming, Journal of Optimization Theory and Applications, Vol. 67, pp. 175–184, 1990.

    Google Scholar 

  25. Penot, J. P.,Preservation of Persistence and Stability under Intersections and Operations, Part 2: Stability, Journal of Optimization Theory and Applications, Vol. 79, pp. 551–561, 1993.

    Google Scholar 

  26. Penot, J. P.,Preservation of Persistence and Stability under Intersections and Operations, Preprint, University of Pau, 1986.

  27. Berge, C.,Espaces Topologiques, Fonctions Multivoques, Dunod, Paris, France, 1990.

    Google Scholar 

  28. Castaing, C., andValadier, M.,Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin, Germany, 1977.

    Google Scholar 

  29. Klein, E., andThompson, A. C.,Theory of Correspondences, John Wiley, New York, New York, 1984.

    Google Scholar 

  30. Janin, R.,Sur la Dualité et la Sensibilité dans les Problèmes de Programme Mathématique, Thèse d'Etat, University of Paris 6, 1974.

  31. Penot, J. P.,On Regularity Conditions in Mathematical Programming, Mathematical Programming Study, Vol. 19, pp. 167–199, 1982.

    Google Scholar 

  32. Robinson, S.,Regularity and Stability for Convex Multivalued Functions, Mathematics of Operations Research, Vol. 1, pp. 130–143, 1976.

    Google Scholar 

  33. Ursescu, C.,Multifunctions with Convex Closed Graph, Czechoslovak Mathematical Journal, Vol. 25, pp. 438–441, 1975.

    Google Scholar 

  34. Rådström, H.,An Embedding Theorem for Spaces of Convex Sets, Proceedings of the American Mathematical Society, Vol. 62, pp. 165–169, 1952.

    Google Scholar 

  35. Penot, J. P.,Convergence of Feasible Sets, Unpublished Manuscript, University of Pau, 1980.

  36. Jameson, G. J. O.,Convex Series, Proceedings of the Cambridge Philosophical Society, Vol. 72, pp. 37–47, 1972.

    Google Scholar 

  37. Tukey, J. W.,Some Notes on the Separation of Convex Sets, Portugalia Mathematica, Vol. 3, pp. 95–102, 1942.

    Google Scholar 

  38. Volle, M.,Quelques Résultats Relatifs à l'Approche par les Tranches de l'Epiconvergence, Preprint, University of Limoges, 1985.

  39. 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, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Germany, Vol. 979, pp. 256–268, 1983.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Applied Mathematics, CNRS-URA 1204, University of Pau, 64000, Pau, France

    J. -P. Penot (Professor)

Authors
  1. J. -P. Penot
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by M. Avriel

Rights and permissions

Reprints and permissions

About this article

Cite this article

Penot, J.P. Preservation of persistence and stability under intersections and operations, part 1: Persistence. J Optim Theory Appl 79, 525–550 (1993). https://doi.org/10.1007/BF00940557

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00940557

Key Words

Search

Navigation