Category:Outer convexification: Difference between revisions

Revision as of 17:54, 20 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 $v^{i}$ 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 [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 $ω (t) \in {0, 1}^{n_{ω}}$ is given by $ω (t) \in [0, 1]^{n_{ω}}$ . We will refer to the two constraints as outer convexification [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

Pages in category "Outer convexification"

This category contains only the following page.

F

F-8 aircraft

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-For time-dependent and space- independent integer controls often another formulation is beneficial, e.g., <bibref>Kirches2010</bibref>. For every element <math>v^i</math> of <math>\Omega</math> a binary control function <math>\omega_i(\cdot)</math> is introduced.
+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
@@ Line 23: / Line 23: @@
 </center>
-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'' <bibref>Sager2005</bibref> 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 <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.
 == References ==