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On the Composition of Convex Envelopes for Quadrilinear Terms

In: Optimization, Simulation, and Control

Author

Listed:
  • Pietro Belotti

    (Clemson University)

  • Sonia Cafieri

    (Ecole Nationale de l’Aviation Civile)

  • Jon Lee

    (University of Michigan)

  • Leo Liberti

    (École Polytechnique)

  • Andrew J. Miller

    (Université Bordeaux 1; RealOpt)

Abstract

Within the framework of the spatial Branch-and-Bound algorithm for solving mixed-integer nonlinear programs, different convex relaxations can be obtained for multilinear terms by applying associativity in different ways. The two groupings $$(({x}_{1}{x}_{2}){x}_{3}){x}_{4}$$ and $$({x}_{1}{x}_{2}{x}_{3}){x}_{4}$$ of a quadrilinear term, for example, give rise to two different convex relaxations. In Cafieri et al. (J Global Optim 47:661–685, 2010) we prove that having fewer groupings of longer terms yields tighter convex relaxations. In this chapter we give an alternative proof of the same fact and perform a computational study to assess the impact of the tightened convex relaxation in a spatial Branch-and-Bound setting.

Suggested Citation

  • Pietro Belotti & Sonia Cafieri & Jon Lee & Leo Liberti & Andrew J. Miller, 2013. "On the Composition of Convex Envelopes for Quadrilinear Terms," Springer Optimization and Its Applications, in: Altannar Chinchuluun & Panos M. Pardalos & Rentsen Enkhbat & E. N. Pistikopoulos (ed.), Optimization, Simulation, and Control, edition 127, pages 1-16, Springer.
    • Handle: RePEc:spr:spochp:978-1-4614-5131-0_1
    DOI: 10.1007/978-1-4614-5131-0_1
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    Citations

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    Cited by:

    1. A. Tsoukalas & A. Mitsos, 2014. "Multivariate McCormick relaxations," Journal of Global Optimization, Springer, vol. 59(2), pages 633-662, July.
    2. Ruth Misener & Christodoulos A. Floudas, 2014. "A Framework for Globally Optimizing Mixed-Integer Signomial Programs," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 905-932, June.
    3. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.

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