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Generic Well-Posedness of Optimization Problems and the Banach-Mazur Game

  • P. S. Kenderov
  • J. P. Revalski
Part of the Mathematics and Its Applications book series (MAIA, volume 331)

Abstract

Let X be a completely regular topological space. Denote, as usual, by C(X) the family of all bounded continuous real-valued functions in X. The space C(X) equipped with the sup-norm ||f|| = sup{| f(x)|: x ∈ X}, f ∈ C(X), becomes a Banach space. Each f ∈ C(X) determines a minimization problem: find x0X with f(x 0) = inf {f(x) : x ∈ X} =: inf (X, f). We designate this problem by (X, f). Among the different properties of the minimization problem (X, f) the following ones are of special interest in the theory of optimization:
  1. (a)

    (X, f) has a solution (existence of the solution);

     
  2. (b)

    the solution set for (X, f) is a singleton (uniqueness of the solution);

     
  3. (c)

    if f(x*) is close to inf (X, f), then x* is a good approximation of the solution of (X, f) (stability of the solution—see bellow the precise definition).

     

Keywords

Topological Space Minimization Problem Solution Mapping Winning Strategy Generic Existence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • P. S. Kenderov
    • 1
  • J. P. Revalski
    • 1
  1. 1.Institute of MathematicsBulgarian Academy of SciencesSofiaBulgaria

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