Category:Outer convexification: Difference between revisions

Revision as of 11:28, 20 November 2010

For time-dependent and space- independent integer controls often another formulation is beneficial, e.g., <bibref>Kirches2010</bibref>. For every element $v^{i}$ of $Ω$ a binary control function $ω_{i} (\cdot)$ is introduced.

The general equation

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

can then be written as

$0 = \sum_{i = 1}^{n_{ω}} F [x, u, v^{i}] ω_{i} (t), t \in [0, t_{f}] .$

If we impose the special ordered set type one condition

$\sum_{i = 1}^{n_{ω}} ω_{i} (t) = 1, t \in [0, t_{f}],$

there is a bijection between every feasible integer function $v (\cdot) \in Ω$ and an appropriately chosen binary function $ω (\cdot) \in {0, 1}^{n_{ω}}$ , compare <bibref>Sager2009</bibref>. The relaxation of $ω (t) \in {0, 1}^{n_{ω}}$ is given by $ω (t) \in [0, 1]^{n_{ω}}$ . We will refer to the two constraints as outer convexification of the original model.

References

Pages in category "Outer convexification"

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F-8 aircraft

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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 ~~{\it~~ outer convexification} of the original model.		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.