Categorical properties of solvability for one class of minimization problems1
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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.
Keywordsdensity Baire category solvability minimization semicontinuity Kadets–Klee property vector optimization
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- 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
- 11.J. Distel, Geometry of Banach Spaces, Springer Verlag (1975).Google Scholar
- 13.J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley (1984).Google Scholar
- 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
- 17.V. V. Semenov, “Typical solvability of some optimal control problems,” Dop. NAN Ukrainy, No. 8, 36–42 (2008).Google Scholar
- 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