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A New Algebraic Geometry Algorithm for Integer Programming

Author

Listed:
  • Dimitris Bertsimas

    (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • Georgia Perakis

    (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • Sridhar Tayur

    (Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213)

Abstract

We propose a new algorithm for solving integer programming (IP) problems that is based on ideas from algebraic geometry. The method provides a natural generalization of the Farkas lemma for IP, leads to a way of performing sensitivity analysis, offers a systematic enumeration of all feasible solutions, and gives structural information of the feasible set of a given IP. We provide several examples that offer insights on the algorithm and its properties.

Suggested Citation

  • Dimitris Bertsimas & Georgia Perakis & Sridhar Tayur, 2000. "A New Algebraic Geometry Algorithm for Integer Programming," Management Science, INFORMS, vol. 46(7), pages 999-1008, July.
    • Handle: RePEc:inm:ormnsc:v:46:y:2000:i:7:p:999-1008
    DOI: 10.1287/mnsc.46.7.999.12033
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    References listed on IDEAS

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    1. Rekha R. Thomas, 1995. "A Geometric Buchberger Algorithm for Integer Programming," Mathematics of Operations Research, INFORMS, vol. 20(4), pages 864-884, November.
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    Cited by:

    1. Hartillo-Hermoso, María Isabel & Jiménez-Tafur, Haydee & Ucha-Enríquez, José María, 2020. "An exact algebraic ϵ-constraint method for bi-objective linear integer programming based on test sets," European Journal of Operational Research, Elsevier, vol. 282(2), pages 453-463.
    2. J. Cole Smith & Churlzu Lim & Aydın Alptekinoğlu, 2009. "New product introduction against a predator: A bilevel mixed‐integer programming approach," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(8), pages 714-729, December.
    3. Castro, F. & Gago, J. & Hartillo, I. & Puerto, J. & Ucha, J.M., 2011. "An algebraic approach to integer portfolio problems," European Journal of Operational Research, Elsevier, vol. 210(3), pages 647-659, May.
    4. Sridhar Tayur, 2017. "OM Forum—An Essay on Operations Management," Manufacturing & Service Operations Management, INFORMS, vol. 19(4), pages 526-533, October.

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