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Optimization over the Pareto front of nonconvex multi-objective optimal control problems

Author

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  • C. Yalçın Kaya

    (University of South Australia)

  • Helmut Maurer

    (Universität Münster)

Abstract

Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.

Suggested Citation

  • C. Yalçın Kaya & Helmut Maurer, 2023. "Optimization over the Pareto front of nonconvex multi-objective optimal control problems," Computational Optimization and Applications, Springer, vol. 86(3), pages 1247-1274, December.
    • Handle: RePEc:spr:coopap:v:86:y:2023:i:3:d:10.1007_s10589-023-00535-7
    DOI: 10.1007/s10589-023-00535-7
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. R. Horst & N. V. Thoai & Y. Yamamoto & D. Zenke, 2007. "On Optimization over the Efficient Set in Linear Multicriteria Programming," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 433-443, September.
    4. Horst, Reiner & Thoai, Nguyen V., 1999. "Maximizing a concave function over the efficient or weakly-efficient set," European Journal of Operational Research, Elsevier, vol. 117(2), pages 239-252, September.
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    Cited by:

    1. William W. Hager & R. Tyrrell Rockafellar & Vladimir M. Veliov, 2023. "Preface to Asen L. Dontchev Memorial Special Issue," Computational Optimization and Applications, Springer, vol. 86(3), pages 795-800, December.
    2. Helmut Maurer & Willi Semmler, 2025. "Multi-objective optimal control with carbon emission and temperature constraints: for achieving a low-fossil-fuel economy," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 33(2), pages 449-471, June.
    3. C. Yalçın Kaya & Lyle Noakes & Erchuan Zhang, 2024. "Multi-objective Variational Curves," Journal of Optimization Theory and Applications, Springer, vol. 201(2), pages 955-975, May.
    4. Braga, Joao Paulo & Chen, Pu & Semmler, Willi, 2025. "Central banks, climate risks, and energy transition—a dynamic macro model and econometric evidence," Macroeconomic Dynamics, Cambridge University Press, vol. 29, pages 1-1, January.

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