Formulating these logical constraint in an ILP

Question

I have these two constraints :

$z \leq My$

$t \leq M'y $

where $z$ and $t$ are two integer variables $ z, t\geq 0$, $y$ is a binary variable, and $M$, $M'$ are two big numbers.

So basically these constraints ensure that if $y = 1$ then $z, t \leq M , M'$ respectively, otherwise $z,t = 0$.

However these constraints won't give me a positive value for $t$ if $z > 0$.

My question is : how to connect variables $z$ and $t$ to ensure that if $z > 0$ then $t > 0$ .

The logical constraint that I want to write is as following:

if $y= 1$ then $z >0$ and $t>0$ .

Thank you.

Answer 1

Let $\epsilon > 0$ be a tolerance for what you consider positive. Now impose linear constraints $z \ge \epsilon y$ and $t \ge \epsilon y$. Because $z$ and $t$ are integer variables, you can take $\epsilon=1$.