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Reverse propagation of McCormick relaxations

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  • Achim Wechsung
  • Joseph Scott
  • Harry Watson
  • Paul Barton

Abstract

Constraint propagation techniques have heavily utilized interval arithmetic while the application of convex and concave relaxations has been mostly restricted to the domain of global optimization. Here, reverse McCormick propagation, a method to construct and improve McCormick relaxations using a directed acyclic graph representation of the constraints, is proposed. In particular, this allows the interpretation of constraints as implicitly defining set-valued mappings between variables, and allows the construction and improvement of relaxations of these mappings. Reverse McCormick propagation yields potentially tighter enclosures of the solutions of constraint satisfaction problems than reverse interval propagation. Ultimately, the relaxations of the objective of a non-convex program can be improved by incorporating information about the constraints. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:63:y:2015:i:1:p:1-36
    DOI: 10.1007/s10898-015-0303-6
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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.
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    4. Joseph Scott & Matthew Stuber & Paul Barton, 2011. "Generalized McCormick relaxations," Journal of Global Optimization, Springer, vol. 51(4), pages 569-606, December.
    5. James E. Falk & Richard M. Soland, 1969. "An Algorithm for Separable Nonconvex Programming Problems," Management Science, INFORMS, vol. 15(9), pages 550-569, May.
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    7. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(6), pages 1195-1198, December.
    8. Agustín Bompadre & Alexander Mitsos, 2012. "Convergence rate of McCormick relaxations," Journal of Global Optimization, Springer, vol. 52(1), pages 1-28, January.
    9. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(2), pages 411-413, April.
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    Cited by:

    1. 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.
    2. Artur M. Schweidtmann & Alexander Mitsos, 2019. "Deterministic Global Optimization with Artificial Neural Networks Embedded," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 925-948, March.
    3. Bo, Yaolong & Xia, Yanghong & Wei, Wei & Li, Zichen & Zhao, Bo & Lv, Zeyan, 2023. "Hyperfine optimal dispatch for integrated energy microgrid considering uncertainty," Applied Energy, Elsevier, vol. 334(C).
    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.
    5. Kamil A. Khan & Harry A. J. Watson & Paul I. Barton, 2017. "Differentiable McCormick relaxations," Journal of Global Optimization, Springer, vol. 67(4), pages 687-729, April.
    6. Jaromił Najman & Alexander Mitsos, 2019. "Tighter McCormick relaxations through subgradient propagation," Journal of Global Optimization, Springer, vol. 75(3), pages 565-593, November.
    7. 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.

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