Computational Approaches for Mixed Integer Optimal Control Problems with Indicator Constraints

  • Michael N. Jung
  • Christian Kirches
  • Sebastian Sager
  • Susanne Sass


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.


Mixed integer optimal control Indicator constraints Vanishing constraints Switched systems MINLP Heavy-duty truck Cruise control Dynamic programming Switching function Partial outer convexification 

Mathematics Subject Classification (2010)

49-04 49M37 65K05 90-08 90C30 90C33 90C39 90C59 90C90 93B40 



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

Funding information

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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Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR)Ruprecht-Karls-Universität HeidelbergHeidelbergGermany
  2. 2.Institut für Mathematische OptimierungTechnische Universität Carolo-Wilhelmina zu BraunschweigBraunschweigGermany
  3. 3.Institut für Mathematische OptimierungOtto-von-Guericke-Universität MagdeburgMagdeburgGermany
  4. 4.Aachener VerfahrenstechnikRheinisch-Westfälische Technische Hochschule AachenAachenGermany

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