Three conditions in the objective function

Answered

Shaojie Liu

August 02, 2023 08:54 Edited

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I am trying to model three conditions in the objective function, which is shown as below. In this function, alpha, beta1, beta2 and m are all constants and the rest of symbols represent variables that are to be determined during the optimization process. I am not sure how to model these three conditions efficiently with Gurobi. Thank you for any help. 

 

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  Shaojie Liu

  August 02, 2023 08:56

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  I noticed three are several posts discussing how to model conditional constraints with Gurobi. But there currently is not a particular example or demonstration how to model three conditions （and even nested conditions）. I hope this post can accomodate such insufficiency. 

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  Lennart Lahrs

  • Gurobi Staff

  August 10, 2023 05:45 Edited

  Hi Shaojie,
  
  You can express this type of objective using auxiliary variables. Please find below some comments and a code example that I hope will help you with your problem.

  • Gurobi can handle bilinear terms, so for each multiplication of more than two variables, we need to define auxiliary variables until we are left with only bilinear terms; here they are called a_squared, term_1, and term_2.
  • To activate and deactivate terms of the objective function, we need conditional variables; here, they are called p_positive and a_positive.
  • Because the third case of your objective function depends on two conditions, I added another auxiliary variable (p_and_a_positive) which is positive if both a and p are positive, using a general constraint

  M = 1e6 # Big M
  E = 1e-6 # Epsilon
  
  p_positive = model.addVars(T, vtype=GRB.BINARY)
  a_positive = model.addVars(T, vtype=GRB.BINARY)
  for t in T:
      model.addConstr(p[t] >= E-M*(1-p_positive[t])) # truly smaller than, using Epsilon
      model.addConstr(p[t] <= M*p_positive[t])
      model.addConstr(a[t] >= E-M*(1-a_positive[t])) # truly smaller than, using Epsilon
      model.addConstr(a[t] <= M*a_positive[t])
  p_and_a_positive = model.addVars(T)
  for t in T:
      model.addGenConstrAnd(p_and_a_positive[t], [p_positive[t], a_positive[t]])
  
  a_squared = model.addVars(T)
  for t in T:
      model.addConstr(a_squared[t] == a[t]**2)
  term_1 = model.addVars(T)
  for t in T:
      model.addConstr(beta_1*p[t] == term_1[t])
  term_2 = model.addVars(T)
  for t in T:
      model.addConstr((beta_2*m*a_squared[t]*v[t])/1000 == term_2[t])
  
  model.setObjective(gp.quicksum(alpha + p_positive[t]*term_1[t] + p_and_a_positive[t]*term_2[t] for t in T))

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  Shaojie Liu

  August 04, 2023 06:47

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  Thank you very much. I think this solution （the code） is very helpful and I can learn a lot from these codes. 

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