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Improved relaxations for the parametric solutions of ODEs using differential inequalities

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  • Joseph Scott
  • Paul Barton

Abstract

A new method is described for computing nonlinear convex and concave relaxations of the solutions of parametric ordinary differential equations (ODEs). Such relaxations enable deterministic global optimization algorithms to be applied to problems with ODEs embedded, which arise in a wide variety of engineering applications. The proposed method computes relaxations as the solutions of an auxiliary system of ODEs, and a method for automatically constructing and numerically solving appropriate auxiliary ODEs is presented. This approach is similar to two existing methods, which are analyzed and shown to have undesirable properties that are avoided by the new method. Two numerical examples demonstrate that these improvements lead to significantly tighter relaxations than previous methods. Copyright Springer Science+Business Media, LLC. 2013

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:57:y:2013:i:1:p:143-176
    DOI: 10.1007/s10898-012-9909-0
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    References listed on IDEAS

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    1. Joseph Scott & Matthew Stuber & Paul Barton, 2011. "Generalized McCormick relaxations," Journal of Global Optimization, Springer, vol. 51(4), pages 569-606, December.
    2. A. B. Singer & P. I. Barton, 2004. "Global Solution of Optimization Problems with Parameter-Embedded Linear Dynamic Systems," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 613-646, June.
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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. 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.
    3. 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.
    4. Spencer D. Schaber & Joseph K. Scott & Paul I. Barton, 2019. "Convergence-order analysis for differential-inequalities-based bounds and relaxations of the solutions of ODEs," Journal of Global Optimization, Springer, vol. 73(1), pages 113-151, January.
    5. 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.
    6. Boris Houska & Benoît Chachuat, 2014. "Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 208-248, July.
    7. Matthew E. Wilhelm & Chenyu Wang & Matthew D. Stuber, 2023. "Convex and concave envelopes of artificial neural network activation functions for deterministic global optimization," Journal of Global Optimization, Springer, vol. 85(3), pages 569-594, March.
    8. Huiyi Cao & Kamil A. Khan, 2023. "General convex relaxations of implicit functions and inverse functions," Journal of Global Optimization, Springer, vol. 86(3), pages 545-572, July.
    9. Kamil A. Khan & Paul I. Barton, 2014. "Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 355-386, November.
    10. Ishan Bajaj & M. M. Faruque Hasan, 2020. "Global dynamic optimization using edge-concave underestimator," Journal of Global Optimization, Springer, vol. 77(3), pages 487-512, July.

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