Category:Outer convexification
From Mintoc
For time-dependent and space- independent integer controls often another formulation is beneficial, e.g., [1]. For every element vi of Ω a binary control function 
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].](/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],](/images/math/0/3/7/037a055b49541ba5af17b1339bd92068.png)
there is a bijection between every feasible integer function 
and an appropriately chosen binary function 
, compare [2]. The relaxation of 
is given by ![\omega(t) \in [0,1]^{n_{\omega}}](/images/math/d/5/7/d574a9fce802239a6bb374d6011b4646.png)
. We will refer to the two constraints as outer convexification [3] of the original model.
References
- ↑ Kirches, C., Sager, S., Bock, H. G., & Schlöder, J. P. (2010). Time-optimal control of automobile test drives with gear shifts. Optimal Control Applications and Methods, 31(2), 137. Bib
- ↑ Sager, S., Reinelt, G., & Bock, H. G. (2009). Direct Methods With Maximal Lower Bound for Mixed-Integer Optimal Control Problems. Mathematical Programming, 118(1), 109. Bib
- ↑ Sager, S. (2005). Numerical methods for mixed–integer optimal control problems. Tönning, Lübeck, Marburg: Der andere Verlag. Bib
Pages in category "Outer convexification"
This category contains only the following page.
