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A practical approach to approximate bilinear functions in mathematical programming problems by using Schur's decomposition and SOS type 2 variables

Author

Listed:
  • S A Gabriel

    (University of Maryland)

  • R García-Bertrand

    (Universidad de Castilla – La Mancha)

  • P Sahakij

    (University of Maryland)

  • A J Conejo

    (Universidad de Castilla – La Mancha)

Abstract

This paper provides a new methodology to solve bilinear, non-convex mathematical programming problems by a suitable transformation of variables. Schur's decomposition and special ordered sets (SOS) type 2 constraints are used resulting in a mixed integer linear or quadratic program in the two applications shown. While Beale, Tomlin and others developed the use of SOS type 2 variables to handle non-convexities, our approach is novel in two aspects. First, the use of Schur's decomposition as an integral part of the approximation step is new and leads to a numerically viable method to separate the variables. Second, the combination of our approach for handling bilinear side constraints in a complementarity or equilibrium problem setting is also new and opens the way to many interesting and realistic modifications to such models. We contrast our approach with other methods for solving bilinear problems also known as indefinite quadratic programs. From a practical point of view our methodology is helpful since no specialized procedures need to be created so that existing solvers can be used. The approach is illustrated with two engineering examples and the mathematical analysis appears in the Appendices.

Suggested Citation

  • S A Gabriel & R García-Bertrand & P Sahakij & A J Conejo, 2006. "A practical approach to approximate bilinear functions in mathematical programming problems by using Schur's decomposition and SOS type 2 variables," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 57(8), pages 995-1004, August.
    • Handle: RePEc:pal:jorsoc:v:57:y:2006:i:8:d:10.1057_palgrave.jors.2602052
    DOI: 10.1057/palgrave.jors.2602052
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    References listed on IDEAS

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    1. Wilson, J. M., 1990. "Generating cuts in integer programming with families of special ordered sets," European Journal of Operational Research, Elsevier, vol. 46(1), pages 101-108, May.
    2. De Wolf, D. & Smeers, Y., 1996. "Optimal dimensioning of pipe networks with application to gas transmission networks," LIDAM Reprints CORE 1249, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Daniel de Wolf & Yves Smeers, 1996. "Optimal Dimensioning of Pipe Networks with Application to Gas Transmission Networks," Operations Research, INFORMS, vol. 44(4), pages 596-608, August.
    4. Richard P. O'Neill & Mark Williard & Bert Wilkins & Ralph Pike, 1979. "A Mathematical Programming Model for Allocation of Natural Gas," Operations Research, INFORMS, vol. 27(5), pages 857-873, October.
    5. Hummeltenberg, Wilhelm, 1984. "Implementations of special ordered sets in MP software," European Journal of Operational Research, Elsevier, vol. 17(1), pages 1-15, July.
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    Cited by:

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    7. E Bustamante-Cedeño & S Arora, 2008. "Stochastic and minimum regret formulations for transmission network expansion planning under uncertainties," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 59(11), pages 1547-1556, November.

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