Originally posted by: JorisK

For 2 integer variables, y=1,...,10, z=1,...,10

Is it possible to express in linear constraints:

NOT(y=3 AND z=7), that is, y cannot be 3 while z=7 and vice versa? I want to extend this to an arbitrary number of integer variables.

I tried the following:

Let x_y3 be a binary variable with the value 1 if and only if y=3, 0 otherwise.

Let x_z7 be a binary variable with the value 1 if and only if z=7, 0 otherwise.

Then the constraint x_y3+x_z7 <= 1 would model the desired behavior.

However, this approach requires that I can force x_y3=1 iff y=3, 0 otherwise. This doesn't seem to be trivial? Any suggestions on how to do this?

In short: expressing a no-good cut in terms of binary variable is easy, but how to express a no-good cut in terms of integer variables?

You can do what you want with two binaries per target value, one for exceeding it and one for falling short of it. I'll demonstrate just for y = 3. Introduce two binary variables u and v, with the constraint u + v <= 1. Constrain y as follows: y <= 3 + 7u - v; y >= 3+u-2v. If u = 1, v = 0 and y is between 4 and 10 (inclusive). If v = 1, u = 0 and y is between 1 and 2 (inclusive). If both u and v are 0, y = 3. Now just plug in (1 - u - v) where you would have had x_y3.