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 ${\displaystyle v^i}$ of ${\displaystyle \mathrm{\Omega }}$ a binary control function ${\displaystyle {\omega }_i(\cdot )}$ 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 ${\displaystyle v(\cdot )\in \mathrm{\Omega }}$ and an appropriately chosen binary function ${\displaystyle \omega (\cdot )\in \{0,1{\}}^{n_{\omega }}}$, 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 ${\displaystyle \omega (t)\in \{0,1{\}}^{n_{\omega }}}$ is given by ${\displaystyle \omega (t)\in [0,1{]}^{n_{\omega }}}$. 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