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A review of computation of mathematically rigorous bounds on optima of linear programs

Author

Listed:
  • Jared T. Guilbeau

    (University of Louisiana)

  • Md. Istiaq Hossain

    (University of Louisiana)

  • Sam D. Karhbet

    (University of Louisiana)

  • Ralph Baker Kearfott

    (University of Louisiana)

  • Temitope S. Sanusi

    (University of Louisiana)

  • Lihong Zhao

    (University of Louisiana)

Abstract

Linear program solvers sometimes fail to find a good approximation to the optimum value, without indicating possible failure. However, it may be important to know how close the value such solvers return is to an actual optimum, or even to obtain mathematically rigorous bounds on the optimum. In a seminal 2004 paper, Neumaier and Shcherbina, propose a method by which such rigorous lower bounds can be computed; we now have significant experience with this method. Here, we review the technique. We point out typographical errors in two formulas in the original publication, and illustrate their impact. Separately, implementers and practitioners can also easily make errors. To help implementers avoid such problems, we cite a technical report where we explain things to mind and where we present rigorous bounds corresponding to alternative formulations of the linear program.

Suggested Citation

  • Jared T. Guilbeau & Md. Istiaq Hossain & Sam D. Karhbet & Ralph Baker Kearfott & Temitope S. Sanusi & Lihong Zhao, 2017. "A review of computation of mathematically rigorous bounds on optima of linear programs," Journal of Global Optimization, Springer, vol. 68(3), pages 677-683, July.
    • Handle: RePEc:spr:jglopt:v:68:y:2017:i:3:d:10.1007_s10898-016-0489-2
    DOI: 10.1007/s10898-016-0489-2
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    References listed on IDEAS

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    1. William Cook & Sanjeeb Dash & Ricardo Fukasawa & Marcos Goycoolea, 2009. "Numerically Safe Gomory Mixed-Integer Cuts," INFORMS Journal on Computing, INFORMS, vol. 21(4), pages 641-649, November.
    2. Ralph Kearfott & Jessie Castille & Gaurav Tyagi, 2013. "A general framework for convexity analysis in deterministic global optimization," Journal of Global Optimization, Springer, vol. 56(3), pages 765-785, July.
    3. Ferenc Domes & Arnold Neumaier, 2012. "Rigorous filtering using linear relaxations," Journal of Global Optimization, Springer, vol. 53(3), pages 441-473, July.
    4. Leonid Chindelevitch & Jason Trigg & Aviv Regev & Bonnie Berger, 2014. "An exact arithmetic toolbox for a consistent and reproducible structural analysis of metabolic network models," Nature Communications, Nature, vol. 5(1), pages 1-9, December.
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