How to model an special constraint in MILP

[s. sadeghi]

  

  Hi all,

   I want to use Gurobi to solve the below describe. In the below example, all variables are boolean variables.

  if  " x1 + x2 + x3 = 0 or 3"  then " t=0 " ;

  else if " x1 + x2 + x3 = 1 or 2 "  then " t=1 " ;

  May I know is it possible to encode those constraints into integer constraints required by Gurobi?

   Best regards,

     Sadegh

[Michael Winkler]

You could add a quadratic constraint  "x1 + x2 + x3 - 3 h1 - [ t * h2 ] = 0", xi, h1 and t binary, h2 integer in [1,2].

If x1+x2+x3 = 0 it follows h1 = t = 0,

If x1+x2+x3 = 3 it follows h1 = 1, t = 0,

If x1+x2+x3 = 1 it follows h1 = 0, t = 1, h2 = 1

If x1+x2+x3 = 2 it follows h1 = 0, t = 1, h2 = 2

Best,

Michael

[Yaarit Cohen]

Have you tried writing it with Big-M?

Best, 

Yaarit Cohen

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[Michael Winkler]

There is a Big-M, the coefficient in front of h1.

Michael

[s. sadeghi]

Hi,

At first thank you for your answer.

Actually it is important to me to have a small number of variables and the constraint be linear. Do you think it can be possible?

Best regards,

Sadegh

[Silke Horn]

Hi,

You will need some auxiliary variables. How about this:

x1 + x2 + x3 - 3 h1 - t - h3 = 0
- t + h3 <= 0
- h2 + h3 <= -1
t + h2 - h3 <= 2

with h3 binary and the rest as in Michael’s answer, i.e., xi, h1, t binary, h2 in {1,2}.

This is linear and does what you asked for. (I got it by putting Michael’s answer through Gurobi’s presolve. So you could just as well use the quadratic one.)

- Silke