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2

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,...

4

If you write $x=B_1-B_2$, $y=B_3-B_4$, and $z=B_5-B_6$ and supply the nine solutions for which $z=x\cdot y$, $B_1+B_2 \le 1$, $B_3+B_4 \le 1$, and $B_5+B_6 \le 1$, PORTA returns $B_i \ge 0$ and the following seven inequalities: \begin{align} - B_1 - B_3 + B_6 &\le 0 \\ - B_1 - B_4 + B_5 &\le 0 \\ - B_2 ...

1

As far as i know you will have to resort to a bi-level optimzation problem: $\min_{A,B} x$ subject to ($\max_x$ subject to $x\leq A$ $x \leq B$) solve a series of non linear problems where you approximate the $\max$ term by something non-linear (a soft max) and let that converge against $\max$ be fine with an $\frac{1}{m}$ error and turn it into an Mixed ...

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