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.

References

1. inverted indicator constraint in gurobipy, https://stackoverflow.com/questions/50366433/inverted-indicator-constraint-in-gurobipy
2. Transposition, https://en.wikipedia.org/wiki/Transposition_(logic)