How do I convert a constraint with a product of two integer variables to a linear constraint?

Question

I have a constraint of the form: $$\theta \leq a_1x_1 + a_2x_2 + a_3x_1x_2$$

where, $x_1$ and $x_2$ are integer variables with ranges $x_1 \in \{0, m\}$ and $x_2 \in \{0, n\}$.

I would want to eliminate the product $x_1x_2$ to make this constraint linear. While I am aware of the ways to do it for binary or continuous variables, I am not sure if the same could be done for integer variables.

It would be a great help if someone could help me out at this.

Thanks a lot.

Answer 1

How about you use $(m+1)(n+1)$ binary variables $x_k$, where the multiplier on $x_k$ is $a_3 a_k$, and $a_k=ij$, $i \in \{0,m\}, j \in \{0,n\}$? You would need an additional constraint $\sum_k x_k = 1$ so that only one of the $x_k$ binary variables is set.