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Quantifying Double McCormick

Author

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  • Emily Speakman

    (University of Michigan, Ann Arbor, Michigan 48109)

  • Jon Lee

    (University of Michigan, Ann Arbor, Michigan 48109)

Abstract

When using the standard McCormick inequalities twice to convexify trilinear monomials, as is often the practice in modeling and software, there is a choice of which variables to group first. For the important case in which the domain is a nonnegative box, we calculate the volume of the resulting relaxation, as a function of the bounds defining the box. In this manner, we precisely quantify the strength of the different possible relaxations defined by all three groupings, in addition to the trilinear hull itself. As a by-product, we characterize the best double-McCormick relaxation. We wish to emphasize that, in the context of spatial branch and bound for factorable formulations, our results do not only apply to variables in the input formulation. Our results apply to monomials that involve auxiliary variables as well. So, our results apply to the product of any three (possibly complicated) expressions in a formulation.

Suggested Citation

  • Emily Speakman & Jon Lee, 2017. "Quantifying Double McCormick," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 1230-1253, November.
    • Handle: RePEc:inm:ormoor:v:42:y:2017:i:4:p:1230-1253
    DOI: 10.1287/moor.2017.0846
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    References listed on IDEAS

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    1. Xuan-Ha Vu & Hermann Schichl & Djamila Sam-Haroud, 2009. "Interval propagation and search on directed acyclic graphs for numerical constraint solving," Computational Optimization and Applications, Springer, vol. 45(4), pages 499-531, December.
    2. Sonia Cafieri & Jon Lee & Leo Liberti, 2010. "On convex relaxations of quadrilinear terms," Journal of Global Optimization, Springer, vol. 47(4), pages 661-685, August.
    3. Miguel A. Lejeune & François Margot, 2016. "Solving Chance-Constrained Optimization Problems with Stochastic Quadratic Inequalities," Operations Research, INFORMS, vol. 64(4), pages 939-957, August.
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    Cited by:

    1. Jon Lee & Daphne Skipper & Emily Speakman, 2022. "Gaining or losing perspective," Journal of Global Optimization, Springer, vol. 82(4), pages 835-862, April.
    2. Emily Speakman & Jon Lee, 2018. "On branching-point selection for trilinear monomials in spatial branch-and-bound: the hull relaxation," Journal of Global Optimization, Springer, vol. 72(2), pages 129-153, October.

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