IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v70y2018i1d10.1007_s10589-018-9981-6.html
   My bibliography  Save this article

On the convergence of the gradient projection method for convex optimal control problems with bang–bang solutions

Author

Listed:
  • J. Preininger

    (Vienna University of Technology)

  • P. T. Vuong

    (Vienna University of Technology)

Abstract

We revisit the gradient projection method in the framework of nonlinear optimal control problems with bang–bang solutions. We obtain the strong convergence of the iterative sequence of controls and the corresponding trajectories. Moreover, we establish a convergence rate, depending on a constant appearing in the corresponding switching function and prove that this convergence rate estimate is sharp. Some numerical illustrations are reported confirming the theoretical results.

Suggested Citation

  • J. Preininger & P. T. Vuong, 2018. "On the convergence of the gradient projection method for convex optimal control problems with bang–bang solutions," Computational Optimization and Applications, Springer, vol. 70(1), pages 221-238, May.
    • Handle: RePEc:spr:coopap:v:70:y:2018:i:1:d:10.1007_s10589-018-9981-6
    DOI: 10.1007/s10589-018-9981-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-018-9981-6
    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/s10589-018-9981-6?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. Yunda Dong, 2015. "Comments on “The Proximal Point Algorithm Revisited”," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 343-349, July.
    2. Martin Seydenschwanz, 2015. "Convergence results for the discrete regularization of linear-quadratic control problems with bang–bang solutions," Computational Optimization and Applications, Springer, vol. 61(3), pages 731-760, July.
    3. Alt, Walter & Schneider, Christopher & Seydenschwanz, Martin, 2016. "Regularization and implicit Euler discretization of linear-quadratic optimal control problems with bang-bang solutions," Applied Mathematics and Computation, Elsevier, vol. 287, pages 104-124.
    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. Yekini Shehu & Qiao-Li Dong & Lulu Liu & Jen-Chih Yao, 2023. "Alternated inertial subgradient extragradient method for equilibrium problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(1), pages 1-30, April.
    2. Bing Tan & Xiaolong Qin & Jen-Chih Yao, 2022. "Strong convergence of inertial projection and contraction methods for pseudomonotone variational inequalities with applications to optimal control problems," Journal of Global Optimization, Springer, vol. 82(3), pages 523-557, March.
    3. Yekini Shehu & Lulu Liu & Qiao-Li Dong & Jen-Chih Yao, 2022. "A Relaxed Forward-Backward-Forward Algorithm with Alternated Inertial Step: Weak and Linear Convergence," Networks and Spatial Economics, Springer, vol. 22(4), pages 959-990, December.
    4. Mina Montazeri & Hamed Kebriaei & Babak N. Araabi, 2023. "A Tractable Truthful Profit Maximization Mechanism Design with Autonomous Agents," Papers 2302.05677, arXiv.org.

    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. T. Scarinci & V. M. Veliov, 2018. "Higher-order numerical scheme for linear quadratic problems with bang–bang controls," Computational Optimization and Applications, Springer, vol. 69(2), pages 403-422, March.
    2. Walter Alt & Ursula Felgenhauer & Martin Seydenschwanz, 2018. "Euler discretization for a class of nonlinear optimal control problems with control appearing linearly," Computational Optimization and Applications, Springer, vol. 69(3), pages 825-856, April.
    3. Dang Hieu & Pham Ky Anh & Nguyen Hai Ha, 2021. "Regularization Proximal Method for Monotone Variational Inclusions," Networks and Spatial Economics, Springer, vol. 21(4), pages 905-932, December.
    4. Walter Alt & C. Yalçın Kaya & Christopher Schneider, 2016. "Dualization and discretization of linear-quadratic control problems with bang–bang solutions," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 4(1), pages 47-77, February.
    5. Chen, Xin & Yuan, Yue & Yuan, Dongmei & Ge, Xiao, 2024. "Optimal control for both forward and backward discrete-time systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 221(C), pages 298-314.
    6. Nikolaus Daniels, 2018. "Tikhonov regularization of control-constrained optimal control problems," Computational Optimization and Applications, Springer, vol. 70(1), pages 295-320, May.
    7. Alt, Walter & Schneider, Christopher & Seydenschwanz, Martin, 2016. "Regularization and implicit Euler discretization of linear-quadratic optimal control problems with bang-bang solutions," Applied Mathematics and Computation, Elsevier, vol. 287, pages 104-124.
    8. Eike Börgens & Christian Kanzow, 2019. "Regularized Jacobi-type ADMM-methods for a class of separable convex optimization problems in Hilbert spaces," Computational Optimization and Applications, Springer, vol. 73(3), pages 755-790, July.
    9. Dang Van Hieu & Jean Jacques Strodiot & Le Dung Muu, 2020. "An Explicit Extragradient Algorithm for Solving Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 476-503, May.
    10. Liang, Yuling & Zhang, Huaguang & Zhang, Juan & Luo, Yanhong, 2021. "Integral reinforcement learning-based guaranteed cost control for unknown nonlinear systems subject to input constraints and uncertainties," Applied Mathematics and Computation, Elsevier, vol. 408(C).
    11. Dang Hieu & Pham Kim Quy, 2023. "One-Step iterative method for bilevel equilibrium problem in Hilbert space," Journal of Global Optimization, Springer, vol. 85(2), pages 487-510, February.

    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:coopap:v:70:y:2018:i:1:d:10.1007_s10589-018-9981-6. 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.