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Computational Approaches for Mixed Integer Optimal Control Problems with Indicator Constraints

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Abstract

Optimal control problems with mixed integer control functions and logical implications, such as a state-dependent restriction on when a control can be chosen (so-called indicator or vanishing constraints) frequently arise in practice. A prominent example is the optimal cruise control of a truck. As every driver knows, admissible gear choices critically depend on the current velocity. A large variety of approaches has been proposed on how to numerically solve this challenging class of control problems. We present a computational study in which the most relevant of them are compared for a reference model problem, based on the same discretization of the differential equations. This comprehends dynamic programming, implicit formulations of the switching decisions, and a number of explicit reformulations, including mathematical programs with vanishing constraints in function spaces. We survey all of these approaches in a general manner, where several formulations have not been reported in the literature before. We apply them to a benchmark truck cruise control problem and discuss advantages and disadvantages with respect to optimality, feasibility, and stability of the algorithmic procedure, as well as computation time.

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Acknowledgements

The work reported in this article was conducted when S. Sass was with Institut für Mathematische Optimierung, Otto-von-Guericke-Universität Magdeburg.

Funding

This study received funding from Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 314838170, GRK 2297 MathCoRe and Priority Programme 1962 “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization,” grant no. KI1839/1-1; the German Federal Ministry of Education and Research, program “Mathematics for Innovations in Industry and Service,” grants no. 05M17MBA-MoPhaPro, 05M18MBA-MoRENet; and program “IKT 2020: Software Engineering,” grant no. 61210304-ODINE. Dynamic programming results were obtained using an implementation by Alexander Buchner.

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Authors and Affiliations

  1. Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR), Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany

    Michael N. Jung

  2. Institut für Mathematische Optimierung, Technische Universität Carolo-Wilhelmina zu Braunschweig, Universitätsplatz 2, 38106, Braunschweig, Germany

    Christian Kirches

  3. Institut für Mathematische Optimierung, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Sebastian Sager

  4. Aachener Verfahrenstechnik, Rheinisch-Westfälische Technische Hochschule Aachen, Forckenbeckstraße 51, 52074, Aachen, Germany

    Susanne Sass

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  1. Michael N. Jung
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  2. Christian Kirches
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  4. Susanne Sass
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Correspondence to Christian Kirches.

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Dedicated to Hans Georg Bock on the occasion of his 70th birthday.

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Jung, M.N., Kirches, C., Sager, S. et al. Computational Approaches for Mixed Integer Optimal Control Problems with Indicator Constraints. Vietnam J. Math. 46, 1023–1051 (2018). https://doi.org/10.1007/s10013-018-0313-z

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