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Extremal Problems in Banach Spaces

  • Vladimir G. Boltyanski
  • Alexander S. Poznyak
Part of the Systems & Control: Foundations & Applications book series (SCFA)

Abstract

This chapter deals with the extension of the Tent Method to Banach spaces. The Abstract Extremal Problem is formulated as an intersection problem. The subspaces in the general positions are introduced. The necessary condition of the separability of a system of convex cones is derived. The criterion of separability in Hilbert spaces is presented. Then the analog of the Kuhn–Tucker Theorem for Banach spaces is discussed in detail.

Keywords

Hilbert Space Banach Space Convex Body General Position Convex Cone 
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References

  1. Boltyanski, V. (1972a), ‘The separation property for a system of convex cones’, Izv. Akad. Nauk Arm. SSR, Ser. Fiz.-Mat. Nauk 7(4), 250–257 (in Russian). MathSciNetGoogle Scholar
  2. Boltyanski, V. (1972b), ‘A theorem on the intersection of sets’, Izv. Akad. Nauk Arm. SSR, Ser. Fiz.-Mat. Nauk 7(5), 325–333 (in Russian). MathSciNetGoogle Scholar
  3. Boltyanski, V. (1975), ‘The tent method in the theory of extremal problems’, Usp. Mat. Nauk 30, 3–65 (in Russian). Google Scholar
  4. Boltyanski, V., Martini, H., & Soltan, V. (1999), Geometric Methods and Optimization Problems, Kluwer Academic, Dordrecht. CrossRefGoogle Scholar
  5. Kuhn, H., & Tucker, A. (1951), ‘Nonlinear programming’, in Proceedings of the Second Berkeley Symposium on Math. Statistics and Probability, University of California Press, Berkeley, pp. 481–492. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Vladimir G. Boltyanski
    • 1
  • Alexander S. Poznyak
    • 2
  1. 1.CIMATGuanajuatoMexico
  2. 2.Automatic Control DepartmentCINVESTAV-IPNMéxicoMexico

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