Category:Outer convexification

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

The general equation

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

can then be written as

$0 = \sum_{i=1}^{n_{\omega}} F[x,u,v^i] \; \omega_i (t), \;\;\;\; t \in [0, t_f].$

If we impose the special ordered set type one condition

$\sum_{i=1}^{n_{\omega}} \omega_i (t) = 1, \;\;\;\; t \in [0, t_f],$

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