Difference between revisions of "Category:Outer convexification"
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| − | For time-dependent and space- independent integer controls often another formulation is beneficial, e.g., < | + | For time-dependent and space- independent integer controls often another formulation is beneficial, e.g., <bib id="Kirches2010" />. For every element <math>v^i</math> of <math>\Omega</math> a binary control function <math>\omega_i(\cdot)</math> is introduced. |
The general equation | The general equation | ||
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| − | there is a bijection between every feasible integer function <math>v(\cdot) \in \Omega</math> and an appropriately chosen binary function <math>\omega(\cdot) \in \{0,1\}^{n_{\omega}}</math>, compare < | + | there is a bijection between every feasible integer function <math>v(\cdot) \in \Omega</math> and an appropriately chosen binary function <math>\omega(\cdot) \in \{0,1\}^{n_{\omega}}</math>, compare <bib id="Sager2009" />. The relaxation of <math>\omega(t) \in \{0,1\}^{n_{\omega}}</math> is given by <math>\omega(t) \in [0,1]^{n_{\omega}}</math>. We will refer to the two constraints as ''outer convexification'' <bib id="Sager2005" /> of the original model. |
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== References == | == References == | ||
| − | < | + | <biblist /> |
[[Category:Model characterization]] | [[Category:Model characterization]] | ||
Latest revision as of 12:10, 23 January 2016
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
![0 = F[x,u,v(t)]](https://mintoc.de/images/math/d/a/6/da6a3ccbfc2da9b4587a746c69806413.png)
can then be written as
![0 = \sum_{i=1}^{n_{\omega}} F[x,u,v^i] \; \omega_i (t), \;\;\;\; t \in [0, t_f].](https://mintoc.de/images/math/a/1/8/a1806642fed3c7bf47027381e6d65a46.png)
If we impose the special ordered set type one condition
![\sum_{i=1}^{n_{\omega}} \omega_i (t) = 1, \;\;\;\; t \in [0, t_f],](https://mintoc.de/images/math/0/3/7/037a055b49541ba5af17b1339bd92068.png)
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 ![\omega(t) \in [0,1]^{n_{\omega}}](https://mintoc.de/images/math/d/5/7/d574a9fce802239a6bb374d6011b4646.png)
. 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
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Pages in category "Outer convexification"
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