I want to write the following constraint:

Let $z$ be an integer variable such that $0\le z\le M$, 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.


1 Answer 1


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.

  • 1
    $\begingroup$ $z$ is asserted to be integer, so $\epsilon=1$. $\endgroup$
    – prubin
    Commented Aug 15, 2020 at 20:55
  • $\begingroup$ Yep. Thanks, @prubin! $\endgroup$
    – RobPratt
    Commented Aug 15, 2020 at 20:56
  • $\begingroup$ @RobPratt Thank you very much for your answer. You've been a great help. $\endgroup$
    – che
    Commented Aug 15, 2020 at 22:09

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