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Tighter McCormick relaxations through subgradient propagation

Author

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  • Jaromił Najman

    (RWTH Aachen University)

  • Alexander Mitsos

    (RWTH Aachen University)

Abstract

Tight convex and concave relaxations are of high importance in deterministic global optimization. We present a method to tighten relaxations obtained by the McCormick technique. We use the McCormick subgradient propagation (Mitsos et al. in SIAM J Optim 20(2):573–601, 2009) to construct simple affine under- and overestimators of each factor of the original factorable function. Then, we minimize and maximize these affine relaxations in order to obtain possibly improved range bounds for every factor resulting in possibly tighter final McCormick relaxations. We discuss the method and its limitations, in particular the lack of guarantee for improvement. Subsequently, we provide numerical results for benchmark cases found in the MINLPLib2 library and case studies presented in previous works, where the McCormick technique appears to be advantageous, and discuss computational efficiency. We see that the presented algorithm provides a significant improvement in tightness and decrease in computational time, especially in the case studies using the reduced space formulation presented in (Bongartz and Mitsos in J Glob Optim 69:761–796, 2017).

Suggested Citation

  • Jaromił Najman & Alexander Mitsos, 2019. "Tighter McCormick relaxations through subgradient propagation," Journal of Global Optimization, Springer, vol. 75(3), pages 565-593, November.
    • Handle: RePEc:spr:jglopt:v:75:y:2019:i:3:d:10.1007_s10898-019-00791-0
    DOI: 10.1007/s10898-019-00791-0
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    References listed on IDEAS

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    1. Dominik Bongartz & Alexander Mitsos, 2017. "Deterministic global optimization of process flowsheets in a reduced space using McCormick relaxations," Journal of Global Optimization, Springer, vol. 69(4), pages 761-796, December.
    2. 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.
    3. Joseph Scott & Matthew Stuber & Paul Barton, 2011. "Generalized McCormick relaxations," Journal of Global Optimization, Springer, vol. 51(4), pages 569-606, December.
    4. 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.
    5. Ambros M. Gleixner & Timo Berthold & Benjamin Müller & Stefan Weltge, 2017. "Three enhancements for optimization-based bound tightening," Journal of Global Optimization, Springer, vol. 67(4), pages 731-757, April.
    6. Achim Wechsung & Joseph Scott & Harry Watson & Paul Barton, 2015. "Reverse propagation of McCormick relaxations," Journal of Global Optimization, Springer, vol. 63(1), pages 1-36, September.
    7. 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.
    8. A. Tsoukalas & A. Mitsos, 2014. "Multivariate McCormick relaxations," Journal of Global Optimization, Springer, vol. 59(2), pages 633-662, July.
    9. Achim Wechsung & Spencer Schaber & Paul Barton, 2014. "The cluster problem revisited," Journal of Global Optimization, Springer, vol. 58(3), pages 429-438, March.
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    Cited by:

    1. Jaromił Najman & Dominik Bongartz & Alexander Mitsos, 2021. "Linearization of McCormick relaxations and hybridization with the auxiliary variable method," Journal of Global Optimization, Springer, vol. 80(4), pages 731-756, August.
    2. Matthew E. Wilhelm & Matthew D. Stuber, 2023. "Improved Convex and Concave Relaxations of Composite Bilinear Forms," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 174-204, April.
    3. Chrysoula D. Kappatou & Dominik Bongartz & Jaromił Najman & Susanne Sass & Alexander Mitsos, 2022. "Global dynamic optimization with Hammerstein–Wiener models embedded," Journal of Global Optimization, Springer, vol. 84(2), pages 321-347, October.
    4. Jason Ye & Joseph K. Scott, 2023. "Extended McCormick relaxation rules for handling empty arguments representing infeasibility," Journal of Global Optimization, Springer, vol. 87(1), pages 57-95, September.

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