Modelling piecewise function constraints

Answered

Jia Qi Ng

• February 20, 2022 02:59

I have defined the piecewise function and was wondering how to define such constraints where a variable is an element of the piecewise function

Any help would be much appreciated! Thank you

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  Jaromił Najman

  Gurobi Staff

  • February 21, 2022 12:55

  Could you share the details of the piecewise-linear function \(PF(\cdot)\)? It is hard to say anything without knowing much about it.

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  Jia Qi Ng

  • February 22, 2022 00:54

  

  

  

  Hi Jaromil, thank you for the response. The piecewise-linear function is defined as shown, hope this helps to clarify my question, thank you!

  

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  Jaromił Najman

  Gurobi Staff

  • February 22, 2022 10:01
  • Edited

  Hi Jia,

  You can try implementing function \(PF\) as part of your model. This means, that you shouldn't try to see \(PF\) as a separate function but see it as constraints defining \(\delta_j\). This would mean to model

  \[\begin{align}
  \max_{x,\delta,FI} &\,\, - \sum_{j=1}^N w_jx_j - \sum_{j=1}^N \delta_j + \delta_0\\
  \text{s.t.} &\,\, \delta_j = \sum_{t=1}^T \lambda_{t^j} f_{t^j} &\quad \forall \, j=1,\dots,N\\
   &\,\, \lambda_{1^j} \leq z_{1^j} &\quad \forall \, j=1,\dots,N \\
  &\,\, \lambda_{t^j} \leq z_{t^j-1}+ z_{t^j} &\quad \forall \, t^j=2,\dots,T^j-1 \,, \forall\, j=1,\dots,N\\
  &\,\, \lambda_{M^j} \leq z_{T^j-1}+ z_{t^j} &\quad \forall \, j=1,\dots,N\\
  &\,\, \sum_{t=1}^{T^j} \lambda_{t^j} = 1&\quad \forall \, j=1,\dots,N\\
  &\,\, \sum_{t=1}^{T^j-1} z_{t^j} = 1&\quad \forall \, j=1,\dots,N\\
  &\,\, \sum_{t=1}^{T^j}  \lambda_{t^j} s_{t^j} = x_j&\quad \forall \, j=1,\dots,N\\
  &z_{t^j} \in \{0,1\}, \lambda_{t^j}\geq 0 &\quad \forall \, t^j=2,\dots,T^j-1 \,, \forall\, j=1,\dots,N
  \end{align}\]

  In the above you would have to adjust \(f_{t^j},s_{t^j}\) to the inputs \(\hat{x},\hat{y}\).
  I don't think that all indices and constraints are 100% correct in the above but I think it is enough to get the idea of what is meant.

  Best regards, 
  Jaromił

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