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On tightness and anchoring of McCormick and other relaxations

Author

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  • Jaromił Najman

    (RWTH Aachen University, AVT - Aachener Verfahrenstechnik, Process Systems Engineering)

  • Alexander Mitsos

    (RWTH Aachen University, AVT - Aachener Verfahrenstechnik, Process Systems Engineering)

Abstract

We say that a convex relaxation of a function is anchored at a particular point in their domains if the values of the function and the relaxation at this point are equal. The opposite of anchoring is offset, i.e., a positive difference between the function and its convex relaxation values over the entire domain. We present theoretical results supported by theoretical and numerical examples showing that anchoring (at corner points) is a useful property but neither necessary nor sufficient for favorable Hausdorff and pointwise convergence order of a relaxation-based bounding scheme. Next, we investigate the tightness and convergence behavior of McCormick relaxations in specific cases. McCormick relaxations have favorable convergence orders, but a positive offset may still slow down the convergence within a simple branch-and-bound algorithm. We demonstrate that use of tighter underlying interval extensions can help reduce the offset and accelerate convergence.

Suggested Citation

  • Jaromił Najman & Alexander Mitsos, 2019. "On tightness and anchoring of McCormick and other relaxations," Journal of Global Optimization, Springer, vol. 74(4), pages 677-703, August.
    • Handle: RePEc:spr:jglopt:v:74:y:2019:i:4:d:10.1007_s10898-017-0598-6
    DOI: 10.1007/s10898-017-0598-6
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    References listed on IDEAS

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    1. Rohit Kannan & Paul I. Barton, 2017. "The cluster problem in constrained global optimization," Journal of Global Optimization, Springer, vol. 69(3), pages 629-676, November.
    2. Jaromił Najman & Alexander Mitsos, 2016. "Convergence analysis of multivariate McCormick relaxations," Journal of Global Optimization, Springer, vol. 66(4), pages 597-628, December.
    3. Ruth Misener & Christodoulos Floudas, 2014. "ANTIGONE: Algorithms for coNTinuous / Integer Global Optimization of Nonlinear Equations," Journal of Global Optimization, Springer, vol. 59(2), pages 503-526, July.
    4. Achim Wechsung & Spencer Schaber & Paul Barton, 2014. "The cluster problem revisited," Journal of Global Optimization, Springer, vol. 58(3), pages 429-438, March.
    5. Achim Wechsung & Paul Barton, 2014. "Global optimization of bounded factorable functions with discontinuities," Journal of Global Optimization, Springer, vol. 58(1), pages 1-30, January.
    6. Agustín Bompadre & Alexander Mitsos, 2012. "Convergence rate of McCormick relaxations," Journal of Global Optimization, Springer, vol. 52(1), pages 1-28, January.
    7. A. Tsoukalas & A. Mitsos, 2014. "Multivariate McCormick relaxations," Journal of Global Optimization, Springer, vol. 59(2), pages 633-662, July.
    8. Agustín Bompadre & Alexander Mitsos & Benoît Chachuat, 2013. "Convergence analysis of Taylor models and McCormick-Taylor models," Journal of Global Optimization, Springer, vol. 57(1), pages 75-114, September.
    9. Joseph Scott & Matthew Stuber & Paul Barton, 2011. "Generalized McCormick relaxations," Journal of Global Optimization, Springer, vol. 51(4), pages 569-606, December.
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