Defining Variable with Dynamic shape

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Afsoon Alidadi Shamsabadi

2022年11月07日 18:19

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Hello, I have been using Gurobi for my PhD thesis. In my current problem, there is a variable matrix whose dimension is not fixed and is defined according to another optimization variable. for example if I define my variable matrix W_i with dimensions (N*M); M is defined according to number of 1 elements in ith row of another variable matrix A with shape (Q*P). (means that A is being decided by the problem) I want to know if it is possible to solve such problem with Gurobi or not.

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  Simranjit Kaur

  • Gurobi Staff

  2025年11月16日 23:25

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  Jaromił Najman

  • Gurobi Staff

  2022年11月08日 09:03 編集済み

  Hi,

  You will have to formulate this property via additional variables and constraints. You have to define all variables for \(W_i\) as if it would attain its maximum dimension.

  M is defined according to number of 1 elements in ith row of another variable matrix A with shape (Q*P)

  You can define an equality constraint and add an integer variable counting the number of 1 elements in \(A\)

  \[\begin{align*}
  \sum A_{ij} = ones
  \end{align*}\]

  where \(ones\) is a non-negative integer variable.

  You can then add a binary variable \(b\) and an inequality constraint for each variable \(x\) of matrix \(W_i\) stating

  \[\begin{align*}
  c\cdot b \leq ones
  \end{align*}\]

  where \(c\) is an integer constant large enough to force \(b=0\) if \(ones\) is not big enough. For example, if variable \(x\) of the \(W_i\) matrix is not present when there are less than 5 ones in \(A\) then \(5 b \leq ones\) will force \(b=0\) for \(ones < 5\). You can then use variable \(b\) to turn on and off variable bounds for variable \(x\) via \(x^{LB} b \leq x \leq x^{UB} b\), with \(x \in [x^{LB}, x^{UB}]\).

  Note that the above is just an example and you will probably have to adjust it for your application. But I think it should be enough to get the idea of how such a formulation can be done.

  Also note that the above idea can probably be improved and maybe there are more involved formulations for your application so it is best to also search for it in the literature.

  Best regards, 
  Jaromił

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