IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v57y2014i3p685-702.html
   My bibliography  Save this article

A numerical method for nonconvex multi-objective optimal control problems

Author

Listed:
  • C. Kaya
  • Helmut Maurer

Abstract

A numerical method is proposed for constructing an approximation of the Pareto front of nonconvex multi-objective optimal control problems. First, a suitable scalarization technique is employed for the multi-objective optimal control problem. Then by using a grid of scalarization parameter values, i.e., a grid of weights, a sequence of single-objective optimal control problems are solved to obtain points which are spread over the Pareto front. The technique is illustrated on problems involving tumor anti-angiogenesis and a fed-batch bioreactor, which exhibit bang–bang, singular and boundary types of optimal control. We illustrate that the Bolza form, the traditional scalarization in optimal control, fails to represent all the compromise, i.e., Pareto optimal, solutions. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • C. Kaya & Helmut Maurer, 2014. "A numerical method for nonconvex multi-objective optimal control problems," Computational Optimization and Applications, Springer, vol. 57(3), pages 685-702, April.
    • Handle: RePEc:spr:coopap:v:57:y:2014:i:3:p:685-702
    DOI: 10.1007/s10589-013-9603-2
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10589-013-9603-2
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10589-013-9603-2?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. L. Grosset & B. Viscolani, 2006. "Reciprocal Optimal Control Problems and the Associated Pareto Frontier," Journal of Optimization Theory and Applications, Springer, vol. 130(1), pages 113-123, July.
    2. Nahid Banihashemi & C. Yalçın Kaya, 2013. "Inexact Restoration for Euler Discretization of Box-Constrained Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 726-760, March.
    3. C.Y. Kaya & J.L. Noakes, 2003. "Computational Method for Time-Optimal Switching Control," Journal of Optimization Theory and Applications, Springer, vol. 117(1), pages 69-92, April.
    4. Henri Bonnel & C. Yalçın Kaya, 2010. "Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 147(1), pages 93-112, October.
    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. Cui, Yunfei & Geng, Zhiqiang & Zhu, Qunxiong & Han, Yongming, 2017. "Review: Multi-objective optimization methods and application in energy saving," Energy, Elsevier, vol. 125(C), pages 681-704.
    2. Stephan Dempe & Gabriele Eichfelder & Jörg Fliege, 2015. "On the effects of combining objectives in multi-objective optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(1), pages 1-18, August.
    3. C. Yalçın Kaya, 2017. "Markov–Dubins path via optimal control theory," Computational Optimization and Applications, Springer, vol. 68(3), pages 719-747, December.
    4. C. Yalçın Kaya, 2019. "Markov–Dubins interpolating curves," Computational Optimization and Applications, Springer, vol. 73(2), pages 647-677, June.
    5. Andreas Lichtenberger & Joao Paulo Braga & Willi Semmler, 2022. "Green Bonds for the Transition to a Low-Carbon Economy," Econometrics, MDPI, vol. 10(1), pages 1-31, March.
    6. Owen Davis & Siavash Radpour, 2022. "Older Workers’ Wages Are Growing—But Not Fast Enough," SCEPA publication series. 2022-02, Schwartz Center for Economic Policy Analysis (SCEPA), The New School.
    7. Andreas Lichtenberger & Joao Paulo Braga & Willi Semmler, 2022. "Green Bonds for the Transition to a Low-Carbon Economy," SCEPA working paper series. 2022-02, Schwartz Center for Economic Policy Analysis (SCEPA), The New School.
    8. Regina S. Burachik & Alexander C. Kalloniatis & C. Yalçın Kaya, 2021. "Sparse Network Optimization for Synchronization," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 229-251, October.
    9. Nguyen Thi Toan & Le Quang Thuy, 2023. "S-Derivative of the Extremum Multifunction to a Multi-objective Parametric Discrete Optimal Control Problem," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 240-265, January.
    10. C. Yalçın Kaya, 2020. "Optimal Control of the Double Integrator with Minimum Total Variation," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 966-981, June.
    11. Nguyen Thi Toan & Le Quang Thuy & Nguyen Tuyen & Yi-Bin Xiao, 2021. "Second-order KKT optimality conditions for multiobjective discrete optimal control problems," Journal of Global Optimization, Springer, vol. 79(1), pages 203-231, January.
    12. Bui Trong Kien & Trinh Duy Binh, 2023. "On the second-order optimality conditions for multi-objective optimal control problems with mixed pointwise constraints," Journal of Global Optimization, Springer, vol. 85(1), pages 155-183, January.

    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. C. Y. Kaya & J. M. Martínez, 2007. "Euler Discretization and Inexact Restoration for Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 134(2), pages 191-206, August.
    2. Elisha R. Pager & Anil V. Rao, 2022. "Method for solving bang-bang and singular optimal control problems using adaptive Radau collocation," Computational Optimization and Applications, Springer, vol. 81(3), pages 857-887, April.
    3. Regina S. Burachik & Alexander C. Kalloniatis & C. Yalçın Kaya, 2021. "Sparse Network Optimization for Synchronization," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 229-251, October.
    4. Michael McAsey & Libin Mou & Weimin Han, 2012. "Convergence of the forward-backward sweep method in optimal control," Computational Optimization and Applications, Springer, vol. 53(1), pages 207-226, September.
    5. R. S. Burachik & C. Y. Kaya & M. M. Rizvi, 2014. "A New Scalarization Technique to Approximate Pareto Fronts of Problems with Disconnected Feasible Sets," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 428-446, August.
    6. C. Yalçın Kaya, 2017. "Markov–Dubins path via optimal control theory," Computational Optimization and Applications, Springer, vol. 68(3), pages 719-747, December.
    7. Haijun Liu & Neng Fan & Panos M. Pardalos, 2012. "Generalized Lagrange Function and Generalized Weak Saddle Points for a Class of Multiobjective Fractional Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 370-381, August.
    8. Xiang Wu & Kanjian Zhang & Changyin Sun, 2013. "Parameter Tuning of Multi-Proportional-Integral-Derivative Controllers Based on Optimal Switching Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 454-472, November.
    9. Henri Bonnel & Christopher Schneider, 2019. "Post-Pareto Analysis and a New Algorithm for the Optimal Parameter Tuning of the Elastic Net," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 993-1027, December.
    10. Kaya, C. Yalcin, 2004. "Time-optimal switching control for the US cocaine epidemic," Socio-Economic Planning Sciences, Elsevier, vol. 38(1), pages 57-72, March.
    11. Yi Jiang & Yi He & Jie Sun, 2011. "Subdifferential properties of the minimal time function of linear control systems," Journal of Global Optimization, Springer, vol. 51(3), pages 395-412, November.
    12. E. G. Birgin & L. F. Bueno & J. M. Martínez, 2016. "Sequential equality-constrained optimization for nonlinear programming," Computational Optimization and Applications, Springer, vol. 65(3), pages 699-721, December.
    13. Henri Bonnel & Léonard Todjihoundé & Constantin Udrişte, 2015. "Semivectorial Bilevel Optimization on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 464-486, November.
    14. Wenhui Luo & Xuewen Tan & Xiufen Zou & Qing Tan, 2023. "Optimal Treatment of Prostate Cancer Based on State Constraint," Mathematics, MDPI, vol. 11(19), pages 1-17, September.
    15. Qun Lin & Ryan Loxton & Kok Teo & Yong Wu, 2015. "Optimal control problems with stopping constraints," Journal of Global Optimization, Springer, vol. 63(4), pages 835-861, December.
    16. Regina S. Burachik & C. Yalçın Kaya & Walaa M. Moursi, 2024. "Infeasible and Critically Feasible Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1219-1245, November.
    17. C. Yalçın Kaya, 2020. "Optimal Control of the Double Integrator with Minimum Total Variation," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 966-981, June.
    18. K. H. Wong & H. W. J. Lee & C. K. Chan & C. Myburgh, 2013. "Control Parametrization and Finite Element Method for Controlling Multi-species Reactive Transport in an Underground Channel," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 168-187, April.
    19. G. Vossen, 2010. "Switching Time Optimization for Bang-Bang and Singular Controls," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 409-429, February.
    20. Nahid Banihashemi & C. Yalçın Kaya, 2013. "Inexact Restoration for Euler Discretization of Box-Constrained Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 726-760, March.

    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:57:y:2014:i:3:p:685-702. 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.