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Linear Optimization with Box Constraints in Banach Spaces

Author

Listed:
  • K. C. Sivakumar

    (Indian Institute of Technology Madras)

  • J. M. Swarna

    (Anna University)

Abstract

Let X be a partially ordered real Banach space, let a,b∈X with a≤b. Let φ be a bounded linear functional on X. We say that X satisfies the box-optimization property (or X is a BOP space) if the box-constrained linear program: max 〈φ,x〉, s.t. a≤x≤b, has an optimal solution for any φ,a and b. Such problems arise naturally in solving a class of problems known as interval linear programs. BOP spaces were introduced (in a different language) and systematically studied in the first author’s doctoral thesis. In this paper, we identify new classes of Banach spaces that are BOP spaces. We present also sufficient conditions under which answers are in the affirmative for the following questions: (i) When is a closed subspace of a BOP space a BOP space? (ii) When is the range of a bounded linear map a BOP space? (iii) Is the quotient space of a BOP space a BOP space?

Suggested Citation

  • K. C. Sivakumar & J. M. Swarna, 2009. "Linear Optimization with Box Constraints in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 377-387, May.
    • Handle: RePEc:spr:joptap:v:141:y:2009:i:2:d:10.1007_s10957-008-9481-4
    DOI: 10.1007/s10957-008-9481-4
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    References listed on IDEAS

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    1. F. E. Udwadia & R. E. Kalaba, 1999. "General Forms for the Recursive Determination of Generalized Inverses: Unified Approach," Journal of Optimization Theory and Applications, Springer, vol. 101(3), pages 509-521, June.
    2. S. Zlobec & A. Ben-Israel, 1973. "Technical Note—Explicit Solutions of Interval Linear Programs," Operations Research, INFORMS, vol. 21(1), pages 390-393, February.
    3. F. E. Udwadia & R. E. Kalaba, 1997. "An Alternative Proof of the Greville Formula," Journal of Optimization Theory and Applications, Springer, vol. 94(1), pages 23-28, July.
    4. F.E. Udwadia & R.E. Kalaba, 2003. "Sequential Determination of the {1, 4}-Inverse of a Matrix," Journal of Optimization Theory and Applications, Springer, vol. 117(1), pages 1-7, April.
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