Difference between revisions of "Category:Outer convexification"
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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 <bibref>Sager2009</bibref>. 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 | + | 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 <bibref>Sager2009</bibref>. 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'' of the original model. |
Revision as of 12:28, 20 November 2010
For time-dependent and space- independent integer controls often another formulation is beneficial, e.g., <bibref>Kirches2010</bibref>. 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 <bibref>Sager2009</bibref>. 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 of the original model.
References
<bibreferences/>
Pages in category "Outer convexification"
This category contains only the following page.
