Category:Outer convexification

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

The general equation

can then be written as

If we impose the special ordered set type one condition

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

References

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