IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v156y2013i3d10.1007_s10957-012-0149-8.html
   My bibliography  Save this article

Convex and Concave Relaxations for the Parametric Solutions of Semi-explicit Index-One Differential-Algebraic Equations

Author

Listed:
  • Joseph K. Scott

    (MIT)

  • Paul I. Barton

    (MIT)

Abstract

A method is presented for computing convex and concave relaxations of the parametric solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). These relaxations are central to the development of a deterministic global optimization algorithm for problems with DAEs embedded. The proposed method uses relaxations of the DAE equations to derive an auxiliary system of DAEs, the solutions of which are proven to provide the desired relaxations. The entire procedure is fully automatable.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:156:y:2013:i:3:d:10.1007_s10957-012-0149-8
    DOI: 10.1007/s10957-012-0149-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-012-0149-8
    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/s10957-012-0149-8?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. Agustín Bompadre & Alexander Mitsos, 2012. "Convergence rate of McCormick relaxations," Journal of Global Optimization, Springer, vol. 52(1), pages 1-28, January.
    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.
    3. Joseph Scott & Matthew Stuber & Paul Barton, 2011. "Generalized McCormick relaxations," Journal of Global Optimization, Springer, vol. 51(4), pages 569-606, December.
    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. 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.
    2. 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.
    3. 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.
    4. 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.

    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. Mario Villanueva & Boris Houska & Benoît Chachuat, 2015. "Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs," Journal of Global Optimization, Springer, vol. 62(3), pages 575-613, July.
    2. A. Tsoukalas & A. Mitsos, 2014. "Multivariate McCormick relaxations," Journal of Global Optimization, Springer, vol. 59(2), pages 633-662, July.
    3. 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.
    4. 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.
    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. 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.
    7. Jaromił Najman & Alexander Mitsos, 2019. "On tightness and anchoring of McCormick and other relaxations," Journal of Global Optimization, Springer, vol. 74(4), pages 677-703, August.
    8. 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.
    9. Achim Wechsung & Spencer Schaber & Paul Barton, 2014. "The cluster problem revisited," Journal of Global Optimization, Springer, vol. 58(3), pages 429-438, March.
    10. 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.
    11. 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.
    12. 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.
    13. Jaromił Najman & Alexander Mitsos, 2016. "Convergence analysis of multivariate McCormick relaxations," Journal of Global Optimization, Springer, vol. 66(4), pages 597-628, December.
    14. 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.
    15. A. Skjäl & T. Westerlund & R. Misener & C. A. Floudas, 2012. "A Generalization of the Classical αBB Convex Underestimation via Diagonal and Nondiagonal Quadratic Terms," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 462-490, August.
    16. N. Kazazakis & C. S. Adjiman, 2018. "Arbitrarily tight $$\alpha $$ α BB underestimators of general non-linear functions over sub-optimal domains," Journal of Global Optimization, Springer, vol. 71(4), pages 815-844, August.
    17. Jai Rajyaguru & Mario E. Villanueva & Boris Houska & Benoît Chachuat, 2017. "Chebyshev model arithmetic for factorable functions," Journal of Global Optimization, Springer, vol. 68(2), pages 413-438, June.
    18. 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.
    19. Joseph Scott & Matthew Stuber & Paul Barton, 2011. "Generalized McCormick relaxations," Journal of Global Optimization, Springer, vol. 51(4), pages 569-606, December.
    20. Polyxeni-Margarita Kleniati & Claire Adjiman, 2014. "Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part I: Theoretical development," Journal of Global Optimization, Springer, vol. 60(3), pages 425-458, November.

    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:156:y:2013:i:3:d:10.1007_s10957-012-0149-8. 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.