## Wednesday, May 16, 2018

### Indicator constraints

In [1] a question is posed about how to implement the implication:$x+y\le 5 \Rightarrow \delta = 1$ Here $$\delta \in \{0,1\}$$ is a binary variable. Indicator constraints are of the form:$\delta = 0 \Rightarrow \text{constraint}$ or $\delta = 1 \Rightarrow \text{constraint}$ The binary variable $$\delta$$ is sometimes called an indicator variable.

From propositional logic we have: \begin{align} & A \Rightarrow B \\ & \Leftrightarrow \\ & \neg B \Rightarrow \neg A\end{align} This is called transposition [2].

From this, we can formulate:$\delta = 0 \Rightarrow x+y\gt 5$ If $$x$$ or $$y$$ is a continuous variable we can choose to write $\delta = 0 \Rightarrow x+y\ge 5.001$ or just $\delta = 0 \Rightarrow x+y\ge 5$ In the latter case we keep things ambiguous for $$x+y=5$$ which is in practice often a good choice.

If  $$x$$ and $$y$$ are integers, we can do: $\delta = 0 \Rightarrow x+y\ge 6$

This trick is quite useful to know when dealing with indicator constraints.