# Egerstedt standard problem

Egerstedt standard problem | |
---|---|

State dimension: | 1 |

Differential states: | 3 |

Discrete control functions: | 3 |

Path constraints: | 1 |

Interior point equalities: | 3 |

The **Egerstedt standard problem** is the problem is of an academic nature and was proposed by Egerestedt to highlight the features of an Hybrid System algorithm in 2006 [Egerstedt2006]**Author: ** *M. Egerstedt; Y. Wardi; H. Axelsson***Journal: ** *IEEE Transactions on Automatic Control***Pages: ** *110--115***Title: ** *Transition-time optimization for switched-mode dynamical systems***Volume: ** *51***Year: ** *2006*

. It has been used since then in many MIOCP research studies (e.g. [Jung2013]**Author: ** *M. Jung; C. Kirches; S. Sager***Booktitle: ** *Facets of Combinatorial Optimization -- Festschrift for Martin Gr\"otschel***Editor: ** *M. J\"unger and G. Reinelt***Pages: ** *387--417***Publisher: ** *Springer Berlin Heidelberg***Title: ** *On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control***Url: ** *http://www.mathopt.de/PUBLICATIONS/Jung2013.pdf***Year: ** *2013*

) for benchmarking of MIOCP algorithms.

## Mathematical formulation

The mixed-integer optimal control problem after partial outer convexification is given by

for .

## Reference Solutions

If the problem is relaxed, i.e., we demand that be in the continuous interval instead of the binary choice , the optimal solution can be determined by using a direct method such as collocation or Bock's direct multiple shooting method.

The optimal objective value of the relaxed problem with is . The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is . The binary control solution was evaluated in the MIOCP by using a Merit function with additional Lagrange term .

## Source Code

Model description is available in

## References

[Egerstedt2006] | M. Egerstedt; Y. Wardi; H. Axelsson (2006): Transition-time optimization for switched-mode dynamical systems. IEEE Transactions on Automatic Control, 51, 110--115 | |

[Jung2013] | M. Jung; C. Kirches; S. Sager (2013): On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control. Facets of Combinatorial Optimization -- Festschrift for Martin Gr\"otschel |

We present numerical results for a benchmark MIOCP from a previous study [157] with the addition of switching constraints. In its original form, the problem was:

After partial outer convexification with respect to the integer control v, the binary
convexified counterpart problem reads