# Category:Outer convexification

For time-dependent and space- independent integer controls often another formulation is beneficial, e.g., [Kirches2010]**Author: ** *C. Kirches; S. Sager; H.G. Bock; J.P. Schl\"oder***Journal: ** *Optimal Control Applications and Methods***Month: ** *March/April***Number: ** *2***Pages: ** *137--153***Title: ** *Time-optimal control of automobile test drives with gear shifts***Url: ** *http://mathopt.de/PUBLICATIONS/Kirches2010.pdf***Volume: ** *31***Year: ** *2010*

. For every element of a binary control function is introduced.

The general equation

can then be written as

If we impose the special ordered set type one condition

there is a bijection between every feasible integer function and an appropriately chosen binary function , compare [Sager2009]**Author: ** *Sager, S.; Reinelt, G.; Bock, H.G.***Journal: ** *Mathematical Programming***Number: ** *1***Pages: ** *109--149***Title: ** *Direct Methods With Maximal Lower Bound for Mixed-Integer Optimal Control Problems***Url: ** *http://mathopt.de/PUBLICATIONS/Sager2009.pdf***Volume: ** *118***Year: ** *2009*

. The relaxation of is given by . We will refer to the two constraints as *outer convexification* [Sager2005]**Address: ** *Tönning, Lübeck, Marburg***Author: ** *S. Sager***Editor: ** *ISBN 3-89959-416-9***Publisher: ** *Der andere Verlag***Title: ** *Numerical methods for mixed--integer optimal control problems***Url: ** *http://mathopt.de/PUBLICATIONS/Sager2005.pdf***Year: ** *2005*

of the original model.

## References

[Kirches2010] | C. Kirches; S. Sager; H.G. Bock; J.P. Schl\"oder (2010): Time-optimal control of automobile test drives with gear shifts. Optimal Control Applications and Methods, 31, 137--153 | |

[Sager2005] | S. Sager (2005): Numerical methods for mixed--integer optimal control problems. (%edition%). Der andere Verlag, Tönning, Lübeck, Marburg, %pages% | |

[Sager2009] | Sager, S.; Reinelt, G.; Bock, H.G. (2009): Direct Methods With Maximal Lower Bound for Mixed-Integer Optimal Control Problems. Mathematical Programming, 118, 109--149 |

## Pages in category "Outer convexification"

This category contains only the following page.