Logical constraint in ILP

Question

I want to write the following constraint:
Let $z$ be an integer variable such that $0le zle M$, and $t$ be a binary variable where $M$ denotes big-M. The logical constraint is as follows:

if $z leq M$ and $z > 0$ then $t=1$;

if $z=0$ then $t=0$.

Is this $z≤Mt$ sufficient? The $t$ and $z$ variables are not in my objective function but variable $t$ is connected to another variable in the objective function?
Thank you very much, I appreciate your help.

RobPratt · Accepted Answer

The big-M constraint $z le M t$ does enforce $z > 0 implies t = 1$, equivalently its contrapositive $t = 0 implies z = 0$, but not the converse $$z = 0 implies t = 0. tag1$$  To enforce $(1)$, consider its contrapositive $$t = 1 implies z > 0 tag2,$$ which you can enforce via big-M constraint $$epsilon - z le (epsilon - 0)(1 - t),$$ equivalently,
$$z ge epsilon t,$$
where $epsilon > 0$ is a tolerance that represents the smallest value of $z$ that you would consider to be positive.

Logical constraint in ILP

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