How to add such an indicator constraints? if a=b then c<d  else c>d

Answered

Yu yue

May 06, 2022 08:55

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I would like to add this instruction constraint:

if a=b then c<d  else c>d

when a=b → k =1 

a !=b → k =0

I know how to handle such situations

if a<b then c<d else c>d

But I'm confused about how to handle the equation.

Kind regards,

Yu

 

 

 

 

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• Official comment

  

  Simranjit Kaur

  • Gurobi Staff

  November 16, 2025 23:38

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• 

  David Torres Sanchez

  • Gurobi Staff

  May 06, 2022 10:32 Edited

  Hi Yu,
  
  For this purpose, you need several auxiliary variables. Edit: I missed one side of the implication.

  1. A continuous variable to represent the difference \(\texttt{y} =\texttt{a}-\texttt{b}\);
  2. A non-negative variable to represent the absolute value of this difference: \(\texttt{yabs} = |y|\);
  3. And a binary variable to indicate whether \(\texttt{yabs}>=0\).

  In Python, using addGenConstrIndicator (overloaded version) you can write

  from gurobipy import *
  
  m = Model()
  
  a = m.addVar(name="a", obj=10)
  b = m.addVar(name="b", obj=1)
  c = m.addVar(name="c", obj=1)
  d = m.addVar(name="d", obj=10)
  
  # Some auxiliary variables
  x = m.addVar(vtype=GRB.BINARY, name="x")
  y = m.addVar(name="y")  # difference a-b
  yabs = m.addVar(lb=0, name="yabs")  # absolute difference
  
  eps = 1.0e-3
  m.addConstr(y == a - b)
  m.addConstr(yabs == abs_(y))
  m.addConstr((x == 1) >> (yabs == 0))
  m.addConstr((x == 0) >> (yabs >= eps))
  
  m.addConstr((x == 1) >> (c <= d - eps))
  m.addConstr((x == 0) >> (d <= c - eps))
  
  m.optimize()

  Where the \(\texttt{eps}\) is a small number used to write strict inequalities.
  
  Cheers, 
  David

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• 

  Yu yue

  May 06, 2022 14:44

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  Hi David，

  Thank you very much，it worked！

  I have one more question, if a is not a decision variable（a∈[1,10]）， b is a decision variable.

  Can it still be solved in this way?

  Best regards,

  Yu

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  David Torres Sanchez

  • Gurobi Staff

  May 09, 2022 08:32 Edited

  Hi Yu,
  
  This also works! Just replace \(\texttt{a}\) with your value.
   
  Best,
  David

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