Indicator function constraints with three conditions

回答済み

El W

2023年08月21日 04:17

• 
• 

I have an optimization problem using gurobi c++: 

objective: minimize L(u), 

where L(u) = {1/2 - u, if u<=0; 1/2(1-u)^2, if 0<u<1; 0, otherwise. } 

How can I create constraints with this indicator function with binary variables and addGenConstrIndicator()? 

 

Please help, thanks. 

関連

• Indicator Constraints

0

• 

  Jaromił Najman

  • Gurobi Staff

  2023年08月21日 08:22 編集済み

  In your case you could use the following formulation

  \[\begin{align*}
  b_0 = 1 &\rightarrow u \leq 0\\
  b_0 = 1 &\rightarrow w = 0.5 - u\\
  b_1 = 1 &\rightarrow u_{aux} = u\\
  b_1 = 1 &\rightarrow w = 0.5 - u_{aux} + 0.5v\\
  b_1 = 0 &\rightarrow u_{aux} = 0\\
  b_2 = 1 &\rightarrow u \geq 1\\
  b_2 = 1 &\rightarrow w = 0\\
  b_0 + b_1 + b_2 &= 1\\
  v &= u_{aux}^2\\
  b_i &\in \{0,1\}, u_{aux} \in [0,1], v \in [0,1]
  \end{align*}\]

  The constraints with \(\rightarrow\) are indicator constraints. Note that your model is nonconvex, thus you will have to set the NonConvex parameter. Note that the auxiliary variable \{v\} is needed to avoid the product \(u_{aux}^2 \cdot b_1\). Variable \(w\) holds the value of your function \(L(u)\).

  0

サインインしてコメントを残してください。