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Extended McCormick relaxation rules for handling empty arguments representing infeasibility

Author

Listed:
  • Jason Ye

    (Georgia Institute of Technology)

  • Joseph K. Scott

    (Georgia Institute of Technology)

Abstract

McCormick’s relaxation technique is one of the most versatile and commonly used methods for computing the convex relaxations necessary for deterministic global optimization. The core of the method is a set of rules for propagating relaxations through basic arithmetic operations. Computationally, each rule operates on four-tuples describing each input argument in terms of a lower bound value, an upper bound value, a convex relaxation value, and a concave relaxation value. We call such tuples McCormick objects. This paper extends McCormick’s rules to accommodate input objects that are empty (i.e., the convex relaxation value lies above the concave, or both relaxation values lie outside the bounds). Empty McCormick objects provide a natural way to represent infeasibility and are readily generated by McCormick-based domain reduction techniques. The standard McCormick rules are strictly undefined for empty inputs and applying them anyway can yield relaxations that are non-convex/concave on infeasible parts of their domains. In contrast, our extended rules always produce relaxations that are well-defined and convex/concave on their entire domain. This capability has important applications in reduced-space global optimization, global dynamic optimization, and domain reduction.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:jglopt:v:87:y:2023:i:1:d:10.1007_s10898-023-01315-7
    DOI: 10.1007/s10898-023-01315-7
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    References listed on IDEAS

    as
    1. 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.
    2. Agustín Bompadre & Alexander Mitsos, 2012. "Convergence rate of McCormick relaxations," Journal of Global Optimization, Springer, vol. 52(1), pages 1-28, January.
    3. A. Tsoukalas & A. Mitsos, 2014. "Multivariate McCormick relaxations," Journal of Global Optimization, Springer, vol. 59(2), pages 633-662, July.
    4. 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.
    5. Joseph Scott & Paul Barton, 2013. "Improved relaxations for the parametric solutions of ODEs using differential inequalities," Journal of Global Optimization, Springer, vol. 57(1), pages 143-176, September.
    6. Joseph Scott & Matthew Stuber & Paul Barton, 2011. "Generalized McCormick relaxations," Journal of Global Optimization, Springer, vol. 51(4), pages 569-606, December.
    7. 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.
    8. Jaromił Najman & Alexander Mitsos, 2019. "Tighter McCormick relaxations through subgradient propagation," Journal of Global Optimization, Springer, vol. 75(3), pages 565-593, November.
    9. Matthew D. Stuber & Paul I. Barton, 2011. "Robust simulation and design using semi-infinite programs with implicit functions," International Journal of Reliability and Safety, Inderscience Enterprises Ltd, vol. 5(3/4), pages 378-397.
    10. 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.
    11. Joseph K. Scott & Paul I. Barton, 2013. "Convex and Concave Relaxations for the Parametric Solutions of Semi-explicit Index-One Differential-Algebraic Equations," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 617-649, March.
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