Does the abs_() function keep the problem a MILP?

Answered

Carmel Hecht

• February 22, 2023 18:02

Hi, I am trying to model a problem as a MILP and solve it using Gurobi. Given a variable X and a constant N, I'd like to define a variable Y which is equal to abs(X-N). I was having trouble formulating it, and than I discovered the helper abs_() method. I wasn't able to find any details regarding the implementation, specifically whether it may hurt the linearity of the MILP? Thank you

Related to

• Formulation
• General Constraints

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• 

  Jue Xue

  Gurobi Staff

  • February 24, 2023 21:09

  Hi Carmel,

   

  Thank you for participating in Gurobi forum and contacting us.

   

  Using abs_() will not hurt the linearity of your MILP.

   

  If you want to know how this may work internally, I will explain it below. Note: you do not need to add any of the constraints below to your model. You may use the 'abs(.)' construct instead.

   

  Let z be a binary variable and M be a big constant (say, >= 2max{|x|}):

  1. use the following inequalities to ensure z = 1 if x >= 0, and z = 0 if x <= 0:
  • zM >= x >= (z-1)M
  2. Create two new inequalities:
  • (1-z)M >= y-x >= (z-1)M
  • zM >= y+x >= -zM

  Then, the above set of linear inequalities is equivalent to y = |x|. I assume you prefer to validate yourself. 

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