Cybernetics and Systems Analysis

, Volume 47, Issue 1, pp 95–105 | Cite as

Categorical properties of solvability for one class of minimization problems1



A lower semicontinuous functional disturbed by a Minkowski functional of a closed bounded convex neighborhood of zero possessing the Kadets–Klee property is minimized on a closed subset X of a reflexive Banach space E. It is proved that the set of parameters for which the problem has a solution contains a Gδ-subset dense in E \ X. It is shown that the reflexivity condition and the condition of the Kadets–Klee property of the neighborhood cannot be weakened. The application to optimization problems for linear systems with vector performance criteria is considered.


density Baire category solvability minimization semicontinuity Kadets–Klee property vector optimization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. B. Stechkin, “Approximative properties of sets in linear normalized spaces,” Rev. Roum. Math. Pur. et Appl., 8, No. 1, 5–18 (1963).MATHGoogle Scholar
  2. 2.
    S. V. Konyagin, “On the approximative properties of arbitrary closed sets in Banach spaces,” Fundam. Prikl. Matem., 3, No. 4, 379–389 (1997).Google Scholar
  3. 3.
    K.-S. Lau, “Almost Chebyshev subsets in reflexive Banach spaces,” Indiana Univ. Math. J., 27, No. 5, 791–795 (1978).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    S. V. Konyagin, “Approximative properties of closed sets in Banach spaces and characterizations of strongly convex spaces,” DAN SSSR, 251, No. 2, 276–280 (1980).MathSciNetGoogle Scholar
  5. 5.
    F. S. De Blasi and J. Myjak, “On a generalized best approximation problem,” J. Approx. Theory, 94, 54–72 (1998).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    C. Li, “On well posed generalized best approximation problems,” J. Approx. Theory, 107, 96–108 (2000).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    R. X. Ni, “Existence of generalized nearest points,” Taiwanese J. Math., 7, No. 1, 115–128 (2003).MATHMathSciNetGoogle Scholar
  8. 8.
    J. Baranger, “Existence de solutions pour des problemes d’optimisation non convexes,” J. Math. Pures Appl., 52, 377–406 (1973).MathSciNetGoogle Scholar
  9. 9.
    C. Li and L. H. Peng, “Ðorosity of perturbed optimization problems in Banach spaces,” J. Math. Anal. Appl., 324, 751–761 (2006).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    R. X. Ni, “Derivatives of generalized farthest functions and existence of generalized farthest points,” J. Math. Anal. Appl., 316, 642–651 (2006).MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Distel, Geometry of Banach Spaces, Springer Verlag (1975).Google Scholar
  12. 12.
    I. Ekeland and G. Lebourg, “Generic Frechet-differentiability and perturbed optimization problems in Banach spaces,” Trans. Amer. Math. Soc., 224, No. 2, 193–216 (1976).MathSciNetGoogle Scholar
  13. 13.
    J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley (1984).Google Scholar
  14. 14.
    J. Penot and M. Thera, “Semi-continuous mappings in general topology,” Archiv der Math., 38, 158–166 (1982).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    M. G. Krein and M. A. Rutman, “Linear operators leaving invariant a cone in a Banach space,” Uspekhi Mat. Nauk, No. 1, 3–95 (1948) (AMS Transl., No. 26, New York (1950)).Google Scholar
  16. 16.
    V. V. Semenov, “Linear variational principle for convex vector maximization,” Cybern. Syst. Analysis, 43, No. 2, 246–252 (2007).MATHCrossRefGoogle Scholar
  17. 17.
    V. V. Semenov, “Typical solvability of some optimal control problems,” Dop. NAN Ukrainy, No. 8, 36–42 (2008).Google Scholar
  18. 18.
    V. V. Semenov, “On solvability of maximization problems in conjugate spaces,” J. Autom. Inform. Sci., Vol.41, Issue 4, 51–55 (2009).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.Taras Shevchenko National University of KyivKyivUkraine

Personalised recommendations