Technical questions regarding binary variables

Answered

Lorenz Wagner

• May 08, 2024 10:54

Hello, I have a purely technical question about Gurobi. Suppose I have the variable \(y_{ijk} \in [0;1]\) . Now I want to define a binary variable \(x_{ijk}\)  that takes the value 1 if \(y_{ijk}=1\)  and 0 otherwise. The constraint \(y_{ijk}\geq x_{ijk}\) should ensure this. If \(y_{ijk}=1\)  applies, then \(x_{ijk}=1\)  can apply, but if \(y_{ijk}<1\) , then \(x_{ijk}=1\)  can never apply. Does the solver then automatically set \(x_{ijk}=0\) ? And how can I prevent \(x_{ijk}=0\)  from applying in the first case, since the constraint does not exclude this?

Related to

• Indicator Constraints

0

• 

  Jaromił Najman

  Gurobi Staff

  • May 13, 2024 08:43
  • Edited

  Hi Lorenz,

  You could use a small tolerance \(\epsilon = 1e-4\) and model your condition using indicator constraints.

  \[\begin{align*}
  x = 1 &\rightarrow\, y = 1\\
  x = 0 &\rightarrow\, y \leq 1 - \epsilon
  \end{align*}\]

  The above states that if \(x=1\) then also \(y=1\) and if \(x=0\) then \(y<1\).

  Please note that the usage of a small tolerance is required, because you are trying to model a strict inequality, but Gurobi does not support these. It is very important to not choose the tolerance \(\epsilon > 0\) too small. In particular it should always be at least one order of magnitude larger than the FeasibilityTol.

  Best regards, 
  Jaromił

   

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