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I have a constraint of the form:

$a_1*a_2 = b_{12}$

where $a_1$ and $a_2$ are integer variables with ranges $a_1∈[{0,1,2,...,m}]$, $a_2∈[{0,1,2,...,n}]$, and $b_{12}∈[{0,1,2,...,m*n}]$.

I would want to eliminate the product $a_1*a_2$ to make this constraint linear.

If $a_1$ and $a_2$ do not equal to $0$, I can use the $log(a_1*a_2)= log(b_{12})$ to convent it into

$log(a_1) +log(a_2) = log(b_{12})$, and then use the pre-defined discrete set to transform it into a linear constraitn with binary vectors. such as

$x_1= log([0,1,2,...,m]) , x_2= log([0,1,2,...,n])$, and $y= log([0,1,2,...,m*n])$.

$x_1*c1 +x_2*c2 = y*d$ where $c_1, c2, d$ are binary vectors and only one element is 1.

However, as in my model, the feasible set of $a_1$ and $a_2$ has 0 element. The "$log$" trick cannot be used as $log(0)$ has no meaning.

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

Thanks a lot.

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  • $\begingroup$ What is the connection between $\{a_1, a_2\}$ and $\{x_1, x_2\}$? $\endgroup$
    – prubin
    Mar 6, 2019 at 20:25
  • $\begingroup$ I'm sorry. I made a typo. I have correct this issue. $x_1$ and $x_2$ should be $a_1$ and $a_2$. Do you have any idea? Highly appreciate your help. $\endgroup$
    – Quentin
    Mar 7, 2019 at 8:07

1 Answer 1

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If you are willing to use binary expansions, you do not need logs. Using your notation, let $c_{1,0},\dots, c_{1,m}$, $c_{2,0},\dots, c_{2,n}$ and $d_0,\dots, d_{m*n}$ be binary variables, with each set summing to 1. Add the following constraints:$$d_0 \ge c_{1,0}$$$$d_0 \ge c_{2,0}$$and$$d_{j*k}\ge c_{1,j} + c_{2,k} - 1\quad\forall j\ge 1,k\ge 1.$$

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