Category:Outer convexification

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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
Link to Google Scholar. For every element v^i of \Omega a binary control function \omega_i(\cdot) is introduced.

The general equation

0 = F[x,u,v(t)]

can then be written as

0 = \sum_{i=1}^{n_{\omega}} F[x,u,v^i] \; \omega_i (t), \;\;\;\; t \in [0, t_f].

If we impose the special ordered set type one condition

\sum_{i=1}^{n_{\omega}} \omega_i (t) = 1, \;\;\;\; t \in [0, t_f],

there is a bijection between every feasible integer function v(\cdot) \in \Omega and an appropriately chosen binary function \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
Link to Google Scholar. The relaxation of \omega(t) \in \{0,1\}^{n_{\omega}} is given by \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
Link to Google Scholar 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--153Link to Google Scholar
[Sager2005]S. Sager (2005): Numerical methods for mixed--integer optimal control problems. (%edition%). Der andere Verlag, Tönning, Lübeck, Marburg, %pages%Link to Google Scholar
[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--149Link to Google Scholar

Pages in category "Outer convexification"

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