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Relaxation methods for mixed-integer optimal control of partial differential equations

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  • Falk Hante
  • Sebastian Sager

Abstract

We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments. Copyright Springer Science+Business Media New York 2013

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  • Falk Hante & Sebastian Sager, 2013. "Relaxation methods for mixed-integer optimal control of partial differential equations," Computational Optimization and Applications, Springer, vol. 55(1), pages 197-225, May.
    • Handle: RePEc:spr:coopap:v:55:y:2013:i:1:p:197-225
    DOI: 10.1007/s10589-012-9518-3
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    References listed on IDEAS

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    1. 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.
    2. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(5), pages 777-788, October.
    3. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(1), pages 151-160, February.
    4. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(3), pages 427-432, June.
    5. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(4), pages 629-637, August.
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    Cited by:

    1. Christoph Buchheim & Renke Kuhlmann & Christian Meyer, 2018. "Combinatorial optimal control of semilinear elliptic PDEs," Computational Optimization and Applications, Springer, vol. 70(3), pages 641-675, July.
    2. Martin Siebenborn, 2018. "A Shape Optimization Algorithm for Interface Identification Allowing Topological Changes," Journal of Optimization Theory and Applications, Springer, vol. 177(2), pages 306-328, May.
    3. Falk M. Hante & Martin Schmidt, 2019. "Complementarity-based nonlinear programming techniques for optimal mixing in gas networks," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 7(3), pages 299-323, September.
    4. 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.
    5. 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.
    6. Yan-Chao Liang & Jane J. Ye, 2021. "Optimality Conditions and Exact Penalty for Mathematical Programs with Switching Constraints," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 1-31, July.
    7. 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.

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