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Duality in Discrete Programming: II. The Quadratic Case

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  • Egon Balas

    (Carnegie-Mellon University)

Abstract

This paper extends the results of "Duality in Discrete Programming" [1] to the case of quadratic objective functions. The paper is, however, self-contained. A pair of symmetric dual quadratic programs is generalized by constraining some of the variables to belong to arbitrary sets of real numbers. Quadratic all-integer and mixed-integer programs are special cases of these problems. The resulting primal problem is shown, subject to a qualification, to have an optimal solution if and only if the dual has one, and in this case the values of their respective objective functions are equal. The dual of a mixed-integer quadratic program can be formulated as a minimax problem whose quadratic objective function is linear in the integer-constrained variables, and whose linear constraint set does not contain the latter. Based on this approach an algorithm is developed for solving integer and mixed-integer quadratic programs.

Suggested Citation

  • Egon Balas, 1969. "Duality in Discrete Programming: II. The Quadratic Case," Management Science, INFORMS, vol. 16(1), pages 14-32, September.
  • Handle: RePEc:inm:ormnsc:v:16:y:1969:i:1:p:14-32
    DOI: 10.1287/mnsc.16.1.14
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    Cited by:

    1. Djerdjour, Mohamed, 1997. "An enumerative algorithm framework for a class of nonlinear integer programming problems," European Journal of Operational Research, Elsevier, vol. 101(1), pages 104-121, August.

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