# Handling $AND$ and $OR$ cases in MILP?

Suppose I want to have an integer program for handling the cases

1. $$x_1>1\wedge x_2>1\wedge x_3>1\wedge\dots\wedge x_n>1\iff\delta=1$$

2. $$x_1>1\vee x_2>1\vee x_3>1\vee\dots\vee x_n>1\iff\delta=1$$

how many number of integer variables are needed to handle case?

Is it possible at least one of them needs at most a constant number of binary variables?

• What encoding have you tried? How many binary variables do you get? Why is it hard to construct one with less variables? – Discrete lizard Apr 3 at 6:04