Different solutions for different modelling approaches

Answered

Paul-Niklas Kandora

• April 25, 2024 10:00

Hey,

I obtain different solutions for different modelling approaches:

1.) Approach 

factor_obj_c = {
    c: m.addVar(name=f"factor_obj_{c}") for c in C
}
quad_expr_c = {
    c: m.addVar(name=f"quad_expr_{c}") for c in C
}
sum_binary_c = {
    c: m.addVar(name=f"sum_bin_{c}") for c in C
}

m.setObjective((1 / C) * gp.quicksum(factor_obj_c[c] for c in C), GRB.MINIMIZE)

for c in C:
    m.addConstr(
        sum_binary_c[c] == gp.quicksum(z_c_r[c, r] for r in R_c[c])
    )
    m.addQConstr(
        quad_expr_c[c] == gp.quicksum(z_c_r[c, r] * mu_c_r[c, r] for r in R_c[c])
    )
    m.addQConstr(
        quad_expr_c[c] == sum_binary_c[c] * factor_obj_c[c]
    )

Please note here that C is some set and R_c some dict which can be accessed through key c. Also one this I should mention is that for my example C = {1} so that we can formulate quad_expr_c in the following as a one-dimensional decision variable (in the example below we refer to quad_expr). The solution approach above returns the wrong solution compared to the situation where I make the following change:

2.) Approach (correct solution)

factor_obj = {
    c: m.addVar(name=f"factor_obj_{c}") for c in C
}
quad_expr = m.addVar(1)

m.setObjective((1 / C) * gp.quicksum(factor_obj[c]
                                     for c in C), GRB.MINIMIZE)

for c in C:
    m.addConstr(
        quad_expr == gp.quicksum(z_c_r[c, r] * mu_c_r[c, r]
                                    for r in R_c[c])
    )
    m.addConstr(
        gp.quicksum(z_c_r[c, r] for r in R_c[c]) * factor_obj[c] ==
        quad_expr
    )

Very grateful for support here!

Best 

Paul

0

• 

  Jaromił Najman

  Gurobi Staff

  • April 25, 2024 14:36

  Hi Paul,

  Could you please provide a minimal reproducible example of both formulations? You can use the write method to generate human-readable LP files by calling

  model.write("myModel.lp")

  You can open the file \(\texttt{myModel.lp}\) in any standard text editor to analyze whether the formulations who are comparing are indeed equivalent.

  Best regards, 
  Jaromił

  0
• 

  Paul-Niklas Kandora

  • April 25, 2024 15:02

  

  

  Jaromił Najman thanks for the response! I was actually able to solve the problem by myself with introducing a support variable (support_quad_expr):
  
  

  factor_obj = {
      c: m.addVar(name=f"factor_obj_{c}") for c in C
  }
  support_quad_expr = m.addVar(1)
  quad_expr_c = {
      c: m.addVar(name=f"quad_expr_{c}") for c in C
  }
  
  

  m.setObjective((1 / C) * gp.quicksum(factor_obj[c]
                                       for c in C), GRB.MINIMIZE)
  
  for c in C:
      m.addConstr(
          quad_expr_c[c] == gp.quicksum(z_c_r[c, r] * mu_c_r[c, r] for r in R_c[c])
      )
      m.addConstr(
          quad_expr_c[c] == support_quad_expr
      )
      m.addConstr(
          gp.quicksum(z_c_r[c, r] for r in R_c[c]) * factor_obj[c]
          == support_quad_expr
      )

  I assume that gurobi had difficulties to understand the following:
  
  

  quad_expr_c[c] == gp.quicksum(z_c_r[c, r] * mu_c_r[c, r] for r in R_c[c])

  Best
  
  Paul

  0

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