IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v171y2016i1d10.1007_s10957-016-0988-9.html
   My bibliography  Save this article

Generalized Derivatives of Differential–Algebraic Equations

Author

Listed:
  • Peter G. Stechlinski

    (Massachusetts Institute of Technology)

  • Paul I. Barton

    (Massachusetts Institute of Technology)

Abstract

Nonsmooth equation-solving and optimization algorithms which require local sensitivity information are extended to systems with nonsmooth parametric differential–algebraic equations embedded. Nonsmooth differential–algebraic equations refers here to semi-explicit differential–algebraic equations with algebraic equations satisfying local Lipschitz continuity and differential right-hand side functions satisfying Carathéodory-like conditions. Using lexicographic differentiation, an auxiliary nonsmooth differential–algebraic equation system is obtained whose unique solution furnishes the desired parametric sensitivities. More specifically, lexicographic derivatives of solutions of nonsmooth parametric differential–algebraic equations are obtained. Lexicographic derivatives have been shown to be elements of the plenary hull of the Clarke (generalized) Jacobian and thus computationally relevant in the aforementioned algorithms. To accomplish this goal, the lexicographic smoothness of an extended implicit function is proved. Moreover, these generalized derivative elements can be calculated in tractable ways thanks to recent advancements in nonsmooth analysis. Forward sensitivity functions for nonsmooth parametric differential–algebraic equations are therefore characterized, extending the classical sensitivity results for smooth parametric differential–algebraic equations.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:171:y:2016:i:1:d:10.1007_s10957-016-0988-9
    DOI: 10.1007/s10957-016-0988-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-016-0988-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-016-0988-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. NESTEROV, Yu., 2005. "Lexicographic differentiation of nonsmooth functions," LIDAM Reprints CORE 1817, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    3. 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).
    4. Bolte, Jérôme & Le, Tam & Pauwels, Edouard & Silveti-Falls, Antonio, 2022. "Nonsmooth Implicit Differentiation for Machine Learning and Optimization," TSE Working Papers 22-1314, Toulouse School of Economics (TSE).
    5. Vikse, Matias & Watson, Harry A.J. & Kim, Donghoi & Barton, Paul I. & Gundersen, Truls, 2020. "Optimization of a dual mixed refrigerant process using a nonsmooth approach," Energy, Elsevier, vol. 196(C).
    6. 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.
    7. Watson, Harry A.J. & Vikse, Matias & Gundersen, Truls & Barton, Paul I., 2018. "Optimization of single mixed-refrigerant natural gas liquefaction processes described by nondifferentiable models," Energy, Elsevier, vol. 150(C), pages 860-876.
    8. NESTEROV, Yurii, 2011. "Random gradient-free minimization of convex functions," LIDAM Discussion Papers CORE 2011001, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:171:y:2016:i:1:d:10.1007_s10957-016-0988-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.