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Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides

Author

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  • Kamil A. Khan

    (Massachusetts Institute of Technology)

  • Paul I. Barton

    (Massachusetts Institute of Technology)

Abstract

Sensitivity analysis provides useful information for equation-solving, optimization, and post-optimality analysis. However, obtaining useful sensitivity information for systems with nonsmooth dynamic systems embedded is a challenging task. In this article, for any locally Lipschitz continuous mapping between finite-dimensional Euclidean spaces, Nesterov’s lexicographic derivatives are shown to be elements of the plenary hull of the (Clarke) generalized Jacobian whenever they exist. It is argued that in applications, and in several established results in nonsmooth analysis, elements of the plenary hull of the generalized Jacobian of a locally Lipschitz continuous function are no less useful than elements of the generalized Jacobian itself. Directional derivatives and lexicographic derivatives of solutions of parametric ordinary differential equation (ODE) systems are expressed as the unique solutions of corresponding ODE systems, under Carathéodory-style assumptions. Hence, the scope of numerical methods for nonsmooth equation-solving and local optimization is extended to systems with nonsmooth parametric ODEs embedded.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:163:y:2014:i:2:d:10.1007_s10957-014-0539-1
    DOI: 10.1007/s10957-014-0539-1
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    References listed on IDEAS

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    1. 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.
    2. NESTEROV, Yu., 2005. "Lexicographic differentiation of nonsmooth functions," LIDAM Reprints CORE 1817, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Ackley, Matthew & Stechlinski, Peter, 2021. "Lexicographic derivatives of nonsmooth glucose-insulin kinetics under normal and artificial pancreatic responses," Applied Mathematics and Computation, Elsevier, vol. 395(C).
    2. Jose Alberto Gomez & Kai Höffner & Kamil A. Khan & Paul I. Barton, 2018. "Generalized Derivatives of Lexicographic Linear Programs," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 477-501, August.
    3. Peter G. Stechlinski & Paul I. Barton, 2016. "Generalized Derivatives of Differential–Algebraic Equations," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 1-26, October.
    4. Peter Stechlinski, 2020. "Optimization-Constrained Differential Equations with Active Set Changes," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 266-293, October.

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