Properties of Efficient Points Sets and Related Topics

  • Vasile Postolicã
Conference paper
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 455)

Abstract

This research work is devoted to the study of the properties for the efficient points sets in separated locally convex spaces, that is, in the framework vector spaces with unknown geometries. Firstly, we present general new properties which illustrate immediate links between Strong Optimization and Vector Optimization and a dual characterization. Afterwards we mention our main results obtained in this field using supernormal cones and the completeness instead of compactness, specifying in the beginning a coincidence between the corresponding efficient points sets for compact subsets with respect to convex, closed and pointed cones and a special kind of Choquet boundaries. This last result gives emphasis to an important connection between Vector Optimization and Potential Theory, being useful also for future developments of these fields in Mathematics. Finally, we give some considerations concerning best approximation simultaneous and vectorial in H-locally convex spaces.

Keywords

Efficient (Pareto minimal) point supernormal cone Choquet boundary H-locally convex space best approximation 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bahya A.O. - Ensembles coniquement bornés et cônes nucléaires dans les éspaces localement convexes separés. Théses de Doctorat de 3-ème Cycle en Mathématiques. L’école Normale Superieure - Takaddoum Rabat (Maroc), 1989.Google Scholar
  2. 2.
    Bahya A.O.–Etude des cônes nucléaires. Ann. Sci. Math. Québes 15, nr. 2, 1992, 123–134.Google Scholar
  3. 3.
    Borwein J.M., Zhuang D.–Super efficiency in vector optimization. Transactions of the American Mathemathical Society, vol. 338, nr. 1, 1993, 105–122.CrossRefGoogle Scholar
  4. 4.
    Bucur I., Postolicâ V.–A coincidence result between sets of efficient points and Choquet boundaries in separated locally convex spaces. Journ. Optimization, 36, 1996, 231–234.CrossRefGoogle Scholar
  5. 5.
    Choquet G. - Lecture on Analysis 1–3 in “Mathematics Lecture Notes Series”, Benjamin, New York/Amsterdam, 1969.Google Scholar
  6. 6.
    Ha T.X.D. - On the existence of efficient points in locally convex spaces. Research Paper, 1993 (to appear in Journal of Global Optimization, Kluwer Academic Publishers, The Netherlands).Google Scholar
  7. 7.
    Ha T.X.D. - A note on a class of cones ensuring the existence of efficient points in bounded complete sets. Research Paper, 1993 (to appear in Journal Optimization, Gordon and Breach Science Publishers).Google Scholar
  8. 8.
    Isac G. - Points critiques des systèmes dynamiques. Cones nucléaires et optimum de Pareto. Research Report, Royal Military College of St. Jean, Québec, Canada, 1981.Google Scholar
  9. 9.
    Isac G.–Sur l’existence de l’optimum de Pareto. Riv. Mat. Univ. Parma, 4, 9, 1983, 303–325.Google Scholar
  10. 10.
    Isac G.–Supernormal cones and absolute summability. Libertas Mathematica, vol. 5, 1985, 17–32.Google Scholar
  11. 11.
    Isac G.. Postolicà V. - The Best Approximation and Optimization in Locally Convex Spaces. Verlag Peter Lang, Frankfurt am Main. Germany, 1993.Google Scholar
  12. 12.
    Sac G. - Pareto optimization in infinite dimensional spaces: the importance of nuclear cones. Journ. of Math. Anal. And Appl., 182. no. 2, 1994, 393–404.Google Scholar
  13. 13.
    Krasnoselski, M. A. Positive solutions of operator equations. Groningen, Noordhoff, 1964.Google Scholar
  14. 14.
    Luc D.T. - Theory of Vector Optimization. Springer Verlag, 1989.Google Scholar
  15. 15.
    Peressini, A. L. - Ordered Topological Vector Spaces. Harper & Row. New York. 1967.Google Scholar
  16. 16.
    Postolicâ V.–Spline functions in H-locally convex spaces. An. St. Univ. “Al. I. Cuza” Iasi, Romania, 27, 1981, 333–338.Google Scholar
  17. 17.
    Postolica V.–Vectorial optimization programs with multifunctions and duality. Ann. Sci. Math. Québec 10, nr. 1, 1986, 85–102.Google Scholar
  18. 18.
    Postolicâ V.–New existence results for efficient points in locally convex spaces ordered by supernormal cones. Journ. of Global Optimization 3, 1993, 233–242, Kluwer Academic Publishers, The Netherlands.Google Scholar
  19. 19.
    Postolicâ V. - Vectorial optimization in locally convex spaces ordered by supernormal cones and extensions. Research Report at the 4th Workshop of the German Working Group on Decision Theory. Hotel Talblick, Holzhau, Germany, March 14–18, 1994, 17 pp.Google Scholar
  20. 20.
    Postolicâ V.–Properties of Pareto sets in locally convex spaces. Journ. Optimization 34, 1995, 223–229.CrossRefGoogle Scholar
  21. 21.
    Precupanu Th.–Espaces linéaires à semi-normes hilbertiennes. An. St. Univ. “Al. 1. Cuza” Iasi, Romania, 15, 1969, 83–93.Google Scholar
  22. 22.
    Precupanu Th.–Scalar minimax properties in vectorial optimization. International Series of Numerical Mathematics, Birkhäuser Verlag Basel, vol. 107, 1992, 299–306.Google Scholar
  23. 23.
    Sterna–Karwat, A.–On the existence of cone maximal points in real topological linear spaces. Israel Journal of Mathematics, 54, 1, 1986. 33–41.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Vasile Postolicã
    • 1
  1. 1.Department of Mathematical SciencesBacãu State UniversityPiatra NeamtRomânia

Personalised recommendations