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Stability of the geometric Ekeland variational principle: Convex case

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Abstract

In geometric terms, the Ekeland variational principle says that a lower-bounded proper lower-semicontinuous functionf defined on a Banach spaceX has a point (x 0,f(x 0)) in its graph that is maximal in the epigraph off with respect to the cone order determined by the convex coneK λ = {(x, α) ∈X × ℝ:λ ∥x∥ ≤ − α}, where λ is a fixed positive scalar. In this case, we write (x 0,f(x 0))∈λ-extf. Here, we investigate the following question: if (x 0,f(x 0))∈λ-extf, wheref is a convex function, and if 〈f n 〉 is a sequence of convex functions convergent tof in some sense, can (x 0,f(x 0)) be recovered as a limit of a sequence of points taken from λ-extf n ? The convergence notions that we consider are the bounded Hausdorff convergence, Mosco convergence, and slice convergence, a new convergence notion that agrees with the Mosco convergence in the reflexive setting, but which, unlike the Mosco convergence, behaves well without reflexivity.

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References

  1. Ekeland, I., andAubin, J. P.,Applied Nonlinear Analysis, Wiley-Interscience, New York, New York, 1984.

    Google Scholar 

  2. De Figueiredo, D. G.,The Ekeland Variational Principle with Applications and Detours, Tata Institute of Fundamental Research, Springer-Verlag, Berlin, Germany, 1989.

    Google Scholar 

  3. Ekeland, I.,Nonconvex Minimization Problems, Bulletin of the American Mathematical Society, Vol. 1, pp. 443–474, 1979.

    Google Scholar 

  4. Phelps, R.,Convex Functions, Monotone Operators, and Differentiability, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Germany, Vol. 1364, 1989.

    Google Scholar 

  5. Attouch, H., andRiahi, H.,Stability Results for Ekeland's ε-Variational Principle and Cone Extremal Solutions, Mathematics of Operations Research, Vol. 18, pp. 173–201, 1993.

    Google Scholar 

  6. Beer, G.,Topologies on Closed and Closed Convex Sets and the Effros Measurability of Set-Valued Functions, Report No. 2, Séminaire d'Analyse Convèxe Montpellier, 1991.

  7. Beer, G.,The Slice Topology: A Viable Alternative to Mosco Convergence in Nonreflexive Spaces, Journal of Nonlinear Analysis, Vol. 19, pp. 271–290, 1992.

    Google Scholar 

  8. Sonntag, Y., andZalinescu, C.,Set Convergences: An Attempt of Classification, Proceedings of the International Conference on Differential Equations and Control Theory, Iasi, Romania, 1990.

  9. Attouch, H., andBeer, G.,On the Convergence of Subdifferentials of Convex Functions, Archiv der Mathematik, Vol. 60, pp. 389–400, 1993.

    Google Scholar 

  10. Beer, G.,Wijsman Convergence of Convex Sets under Renorming, Journal of Nonlinear Analysis (to appear).

  11. Tychonov, A.,On the Stability of the Functional Optimization Problem, USSR Computational Mathematics and Mathematical Physics, Vol. 6, pp. 28–33, 1966.

    Google Scholar 

  12. Furi, M., andVignoli, A.,About Well-Posed Optimization Problems for Functionals in Metric Spaces, Journal of Optimization Theory and Applications, Vol. 5, pp. 225–229, 1970.

    Google Scholar 

  13. Lucchetti, R.,Some Aspects of the Connections between Hadamard and Tyhonov Well-Posedness of Convex Programs, Bollettino della Unione Matematica Italiana, Serie VI, Analisi Funzionale e Applicazioni, Vol. 1, pp. 337–344, 1982.

    Google Scholar 

  14. Beer, G., andLucchetti, R.,Convex Optimization and the Epidistance Topology, Transactions of the American Mathematical Society, Vol. 327, pp. 795–813, 1991.

    Google Scholar 

  15. Azé, D., andPenot, J. P.,Qualitative Results about the Convergence of Convex Sets and Convex Functions, Optimization and Nonlinear Analysis, Haifa, 1990; Pitman Research Notes in Mathematics Series, Longman, Harlow, Great Britain, Vol. 244, pp. 1–24, 1992.

  16. Joly, J.,Une Famille de Topologies sur l'Ensemble des Fonctions Convexes pour Lesquelles la Polarité est Bicontinue, Journal de Mathématiques Pures et Appliquées, Vol. 52, pp. 421–441, 1973.

    Google Scholar 

  17. Wijsman, R.,Convergence of Sequences of Convex Sets, Cones, and Functions, Part 2, Transactions of the American Mathematical Society, Vol. 123, pp. 32–45, 1966.

    Google Scholar 

  18. Mosco, U.,Convergence of Convex Sets and of Solutions of Variational Inequalities, Advances in Mathematics, Vol. 3, pp. 510–585, 1969.

    Google Scholar 

  19. Mosco, U.,On the Continuity of the Young-Fenchel Transform, Journal of Mathematical Analysis and Applications, Vol. 35, pp. 518–535, 1971.

    Google Scholar 

  20. Attouch, H.,Variational Convergence for Functions and Operators, Pitman, New York, New York, 1984.

    Google Scholar 

  21. Aubin, J. P., andFrankowska, H.,Set-Valued Analysis, Birkhäuser, Boston, Massachusetts, 1990.

    Google Scholar 

  22. Kuratowski, K.,Topology, Vol. 1, Academic Press, New York, New York, 1966.

    Google Scholar 

  23. Beer, G., andPai, D.,On Convergence of Convex Sets and Relative Chebyshev Centers, Journal of Approximation Theory, Vol. 62, pp. 147–169, 1990.

    Google Scholar 

  24. Borwein, J., andFitzpatrick, S.,Mosco Convergence and the Kadec Property, Proceedings of the American Mathematical Society, Vol. 106, pp. 843–852, 1989.

    Google Scholar 

  25. Beer, G.,On Mosco Convergence of Convex Sets, Bulletin of the Australian Mathematical Society, Vol. 38, pp. 239–253, 1988.

    Google Scholar 

  26. Salinetti, G., andWets, R.,On the Relationship between Two Types of Convergence for Convex Functions, Journal of Mathematical Analysis and Applications, Vol. 60, pp. 211–226, 1977.

    Google Scholar 

  27. Sonntag, Y.,Convergence au Sens de Mosco: Théorie et Applications à l'Approximation des Solutions d'Inéquations, Thesis, Université de Provence, Marseille, France, 1982.

    Google Scholar 

  28. Tsukada, M.,Convergence of Best Approximations in a Smooth Banach Space, Journal of Approximation Theory, Vol. 40, pp. 301–309, 1984.

    Google Scholar 

  29. Beer, G., andBorwein, J.,Mosco Convergence and Reflexivity, Proceedings of the American Mathematical Society, Vol. 109, pp. 427–436, 1990.

    Google Scholar 

  30. Attouch, H., andWets, R.,Quantitative Stability of Variational Systems: I. The Epigraphical Distance, Transactions of the American Mathematical Society, Vol. 328, pp. 695–730, 1991.

    Google Scholar 

  31. Beer, G.,Conjugate Convex Functions and the Epidistance Topology, Proceedings of the American Mathematical Society, Vol. 108, pp. 117–128, 1990.

    Google Scholar 

  32. Azé, D., andPenot, J. P.,Operations on Convergent Families of Sets and Functions, Optimization, Vol. 21, pp. 521–534, 1990.

    Google Scholar 

  33. Attouch, H., Lucchetti, R., andWets, R.,The Topology of the ρ-Hausdorff Distance, Annali di Matematica Pura ed Applicata, Vol. 160, pp. 303–320, 1991.

    Google Scholar 

  34. Penot, J. P.,The Cosmic Hausdorff Topology, the Bounded Hausdorff Topology, and Continuity of Polarity, Proceedings of the American Mathematical Society, Vol. 113, pp. 275–286, 1991.

    Google Scholar 

  35. Beer, G., andLucchetti, R.,The Epidistance Topology: Continuity and Stability Results with Applications to Convex Optimization Problems, Mathematics of Operations Research, Vol. 17, pp. 715–726, 1992.

    Google Scholar 

  36. Attouch, H., Ndoutoume, J., andThera, M.,Epigraphical Convergence of Functions and Convergence of Their Derivatives in Banach Spaces, Report No. 9, Séminaire d'Analyse Convèxe Montpellier, 1990.

  37. Beer, G. andBorwein, J.,Mosco and Slice Convergence of Level Sets and Graphs of Linear Functionals, Journal of Mathematical Analysis and Applications, Vol. 175, pp. 53–67, 1993.

    Google Scholar 

  38. Beer, G., andLucchetti, R.,Weak Topologies for the Closed Subsets of a Metrizable Space, Transactions of the American Mathematical Society, Vol. 335, pp. 805–822, 1993.

    Google Scholar 

  39. Georgiev, P.,The Strong Ekeland Variational Principle, the Strong Drop Theorem, and Applications, Journal of Mathematical Analysis and Applications, Vol. 131, pp. 1–21, 1988.

    Google Scholar 

  40. Asplund, E., andRockafellar, R. T.,Gradients of Convex Functions, Transactions of the American Mathematical Society, Vol. 139, pp. 443–467, 1969.

    Google Scholar 

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Authors and Affiliations

  1. Département de Mathématiques, Université Montpellier II, Montpellier, France

    H. Attouch (Professor)

  2. Department of Mathematics, California State University at Los Angeles, Los Angeles, California

    G. Beer (Professor)

  3. Université Montpellier II, Montpellier, France

    G. Beer (Professor)

Authors
  1. H. Attouch
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  2. G. Beer
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Additional information

Communicated by M. Avriel

The research of the second author was partially supported by National Science Foundation Grant No. DMS-90-01096. He wishes to thank the first author and the Université Montpellier II for their hospitality.

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Attouch, H., Beer, G. Stability of the geometric Ekeland variational principle: Convex case. J Optim Theory Appl 81, 1–19 (1994). https://doi.org/10.1007/BF02190310

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