Can integer programming problem with non-continuous feasible region be solved with Gurobi?

Answered

Hyewook Kim

July 14, 2022 04:31

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Hello, I'm looking for a suitable and easy-to-use solver for the specific problem below.

Can the problem be formulated with gurobi solver?

 

Minimize (int) x

s.t.          x - a >= 0 (where a is a scalar given in the problem)

where     feasible region for x is a bounded and non-continuous("jump") integer space such as [1,2,3,4,8,9,10,11,18,19,20].

 

 

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• Official comment

  

  Simranjit Kaur

  • Gurobi Staff

  November 16, 2025 23:32

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  Jonasz Staszek

  • Community Moderator

  July 14, 2022 05:17

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  Hi Hyewook,

  yes, this is possible, with the help of a few indicator constraints and variables. Perhaps this post will give you an idea.

  Best regards,
  Jonasz

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  Hyewook Kim

  July 25, 2022 01:27

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  Thank you for a help, and sorry for a late reply.

  I certainly got an idea from the post above that adding extra binary variables(z) can implement 'Or constraints', such as 'less than n1 Or greater than n2'.

   

  Now, I'm struggling with how to utilize this idea to present choose-one-of-numerous-'And constraints' problem like my original question.

  As written above, x should be between either one of (1<=x<=4) or (8<=x<=11) or (18<=x<=20) ranges.

  And adding binary variable might help the solver to choose one of these 3 between-ranges.

  But I'm still middle of figuring out how to implement 'And constraints' (e.g. greater than 1 and less than 4).

   

  Best regards,

  HW

   

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  Jonasz Staszek

  • Community Moderator

  July 25, 2022 05:18

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  Hi Hyewook,

  the "And" part is governed by adding two indicator constraints for each indicator variable. For example:

  z[i, 1] = m.addVar(vtype="B",name=f"indicator_{i}_1")
  z[i, 2] = m.addVar(vtype="B",name=f"indicator_{i}_2")
  z[i, 3] = m.addVar(vtype="B",name=f"indicator_{i}_3")
      
  m.addLConstr(z[i, 1] + z[i, 2] + z[i, 3] == 1)
      
  m.addGenConstrIndicator(z[i, 1], True, x, gp.GRB.GREATER_EQUAL, 1.0) 
  m.addGenConstrIndicator(z[i, 1], True, x, gp.GRB.LESS_EQUAL, 4.0) 
  
  m.addGenConstrIndicator(z[i, 2], True, x, gp.GRB.GREATER_EQUAL, 8.0) 
  m.addGenConstrIndicator(z[i, 2], True, x, gp.GRB.LESS_EQUAL, 11.0) 
  
  m.addGenConstrIndicator(z[i, 3], True, x, gp.GRB.GREATER_EQUAL, 18.0) 
  m.addGenConstrIndicator(z[i, 3], True, x, gp.GRB.LESS_EQUAL, 20.0) 

  Hope this helps.

  Best regards
  Jonasz

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