Conditional constraint with a strict inequality

Question

It's almost this question: Formulating the conditional constraint

But there they have non-strict inequality. I have $x_i$ a boolean decision var and $Q_i$ as a nonnegative integer decision variable such that

1. if $x_i = 0$, then $Q_i = 0$
2. if $x_i = 1$, then $Q_i \gt 0$ (note the strict inequality!).

Lets say I dont have upper bound on $Q_i$; is there a mathematical relation between $x_i$ and $Q_i$ you can write directly or is the following way to go?

dvar boolean x[I];
dvar int+ Q[I];

subject to
{
  forall(i in I) {
     (x[i]==0) => (Q[i] == 0);
     (x[i]==1) => (Q[i] > 0);
}

I can formulate the constraint other way but does it help I'm not sure:

1. if $Q_i = 0$, then $x_i = 0$
2. if $Q_i \gt 0$, then $x_i = 1$.

Answer 1

As already mentioned in the comments: If $𝑄_i$ is an integer variable, then $𝑄_i > 0$ is equivalent to $𝑄_i \geq 1$. Hence, you can use the indicator constraints offered by the most commercial solvers in case you don't know upper bounds, i.e. (x[i]==1) => (Q[i] >= 1). Note also prubin's comment on how one can calculate an upper bound:

By the way, you can (usually) calculate an upper bound for $Q_i$ by relaxing the integrality constraints and maximizing Qi subject to the constraints. The only two issues here are (1) the LP might be unbounded and (2) if the problem is big enough, solving all those LPs (one for each i) might take too much time.