IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v78y2021i2d10.1007_s10589-020-00244-5.html
   My bibliography  Save this article

On mixed-integer optimal control with constrained total variation of the integer control

Author

Listed:
  • Sebastian Sager

    (Otto-von-Guericke-Universität Magdeburg)

  • Clemens Zeile

    (Otto-von-Guericke-Universität Magdeburg)

Abstract

The combinatorial integral approximation (CIA) decomposition suggests solving mixed-integer optimal control problems by solving one continuous nonlinear control problem and one mixed-integer linear program (MILP). Unrealistic frequent switching can be avoided by adding a constraint on the total variation to the MILP. Within this work, we present a fast heuristic way to solve this CIA problem and investigate in which situations optimality of the constructed feasible solution is guaranteed. In the second part of this article, we show tight bounds on the integrality gap between a relaxed continuous control trajectory and an integer feasible one in the case of two controls. Finally, we present numerical experiments to highlight the proposed algorithm’s advantages in terms of run time and solution quality.

Suggested Citation

  • Sebastian Sager & Clemens Zeile, 2021. "On mixed-integer optimal control with constrained total variation of the integer control," Computational Optimization and Applications, Springer, vol. 78(2), pages 575-623, March.
  • Handle: RePEc:spr:coopap:v:78:y:2021:i:2:d:10.1007_s10589-020-00244-5
    DOI: 10.1007/s10589-020-00244-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-020-00244-5
    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/s10589-020-00244-5?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. H. Axelsson & Y. Wardi & M. Egerstedt & E. I. Verriest, 2008. "Gradient Descent Approach to Optimal Mode Scheduling in Hybrid Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 136(2), pages 167-186, February.
    2. Sebastian Sager & Michael Jung & Christian Kirches, 2011. "Combinatorial integral approximation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(3), pages 363-380, June.
    3. Пигнастый, Олег & Koжевников, Георгий, 2019. "Распределенная Динамическая Pde-Модель Программного Управления Загрузкой Технологического Оборудования Производственной Линии [Distributed dynamic PDE-model of a program control by utilization of t," MPRA Paper 93278, University Library of Munich, Germany, revised 02 Feb 2019.
    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. Marvin Severitt & Paul Manns, 2023. "Efficient Solution of Discrete Subproblems Arising in Integer Optimal Control with Total Variation Regularization," INFORMS Journal on Computing, INFORMS, vol. 35(4), pages 869-885, July.

    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. Sven Leyffer & Paul Manns & Malte Winckler, 2021. "Convergence of sum-up rounding schemes for cloaking problems governed by the Helmholtz equation," Computational Optimization and Applications, Springer, vol. 79(1), pages 193-221, May.
    2. Elisa Alòs & Maria Elvira Mancino & Tai-Ho Wang, 2019. "Volatility and volatility-linked derivatives: estimation, modeling, and pricing," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 321-349, December.
    3. Jose Cruz & Daniel Sevcovic, 2020. "On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models," Papers 2003.03851, arXiv.org.
    4. Ana Werlang & Gabriel Cunha & João Bastos & Juliana Serra & Bruno Barbosa & Luiz Barroso, 2021. "Reliability Metrics for Generation Planning and the Role of Regulation in the Energy Transition: Case Studies of Brazil and Mexico," Energies, MDPI, vol. 14(21), pages 1-27, November.
    5. Marvin Severitt & Paul Manns, 2023. "Efficient Solution of Discrete Subproblems Arising in Integer Optimal Control with Total Variation Regularization," INFORMS Journal on Computing, INFORMS, vol. 35(4), pages 869-885, July.
    6. Elena-Corina Cipu, 2019. "Duality Results in Quasiinvex Variational Control Problems with Curvilinear Integral Functionals," Mathematics, MDPI, vol. 7(9), pages 1-9, September.
    7. Hanno Gottschalk & Marco Reese, 2021. "An Analytical Study in Multi-physics and Multi-criteria Shape Optimization," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 486-512, May.
    8. Karel Van Bockstal, 2020. "Existence of a Unique Weak Solution to a Nonlinear Non-Autonomous Time-Fractional Wave Equation (of Distributed-Order)," Mathematics, MDPI, vol. 8(8), pages 1-16, August.
    9. Assed Haddad & Ahmed Hammad & Danielle Castro & Diego Vasco & Carlos Alberto Pereira Soares, 2021. "Framework for Assessing Urban Energy Sustainability," Sustainability, MDPI, vol. 13(16), pages 1-18, August.
    10. Savin Treanţă, 2019. "On Locally and Globally Optimal Solutions in Scalar Variational Control Problems," Mathematics, MDPI, vol. 7(9), pages 1-8, September.
    11. Darvishi, M.T. & Najafi, Mohammad & Wazwaz, Abdul-Majid, 2021. "Conformable space-time fractional nonlinear (1+1)-dimensional Schrödinger-type models and their traveling wave solutions," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    12. S. Göttlich & A. Potschka & C. Teuber, 2019. "A partial outer convexification approach to control transmission lines," Computational Optimization and Applications, Springer, vol. 72(2), pages 431-456, March.
    13. Julien Hok & Sergei Kucherenko, 2021. "Pricing and Risk Analysis in Hyperbolic Local Volatility Model with Quasi Monte Carlo," Papers 2106.08421, arXiv.org.
    14. Ivan Francisco Yupanqui Tello & Alain Vande Wouwer & Daniel Coutinho, 2021. "A Concise Review of State Estimation Techniques for Partial Differential Equation Systems," Mathematics, MDPI, vol. 9(24), pages 1-15, December.
    15. Christian Klein & Julien Riton & Nikola Stoilov, 2021. "Multi-domain spectral approach for the Hilbert transform on the real line," Partial Differential Equations and Applications, Springer, vol. 2(3), pages 1-19, June.
    16. Marco Cirant & Roberto Gianni & Paola Mannucci, 2020. "Short-Time Existence for a General Backward–Forward Parabolic System Arising from Mean-Field Games," Dynamic Games and Applications, Springer, vol. 10(1), pages 100-119, March.
    17. Song, Xiaona & Wang, Mi & Song, Shuai & Wang, Zhen, 2021. "Observer-based sliding mode control for stochastic hyperbolic PDE systems with quantized output signal," Applied Mathematics and Computation, Elsevier, vol. 393(C).
    18. Mario Abundo & Enrica Pirozzi, 2019. "On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes," Mathematics, MDPI, vol. 7(10), pages 1-12, October.
    19. Christian Kuehn & Cinzia Soresina, 2020. "Numerical continuation for a fast-reaction system and its cross-diffusion limit," Partial Differential Equations and Applications, Springer, vol. 1(2), pages 1-26, April.
    20. Zaiping Zhu & Andres Iglesias & Liqi Zhou & Lihua You & Jianjun Zhang, 2022. "PDE-Based 3D Surface Reconstruction from Multi-View 2D Images," Mathematics, MDPI, vol. 10(4), pages 1-17, 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:78:y:2021:i:2:d:10.1007_s10589-020-00244-5. 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.