SOS Constraint with sum

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Alexander Haberman

2020年06月17日 02:02

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Is there a way to create a SOS constraint where one of the variables is a sum of variables? I have n+1 non-negative variables denoted x_1,...x_n, y and would like to add a constraint that either y=0 or x_1=...=x_n=0. Since the variables are all greater than or equal to zero, it seems like a natural way to achieve this would be to have a type 1 SOS constraint where the two variables are y and x_1+...+x_n. I know this could be achieved by creating n type 1 SOS constraints of the form (y, x_i), but n is very large and this must be done for multiple y's so I'd like to avoid that. 

Thanks!

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  Simranjit Kaur

  • Gurobi Staff

  2025年06月23日 03:11 編集済み

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  Richard Oberdieck

  • Gurobi Staff

  2020年06月17日 07:10

  Hi Alexander,

  I would not use SOS constraints in this case, but big-M. Specifically, let M1 be the big-M value of y and M2 be the big-M value of the sum(x_i). Then I would introduce a binary variable z such that:

  y <= M1z
  sum(x_i) <= M2(1-z)

  Then, if z=1, you have sum(x_i)<=0 which due to the non-negativity means x_i=0 for all x_i. Equally, if z=0, then you have y<=0 which due to the non-negativity means y=0. You can check some slides I made about this here (page 15).

   

  Hope this helps

  Richard

   

   

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  Alexander Haberman

  2020年06月26日 14:29

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  Thanks Richard, that is very helpful!

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