How to raise the power of a variable?

Answered

BISWAJIT KAR

• February 03, 2022 12:24

I have to solve a Non linear problem, with one constraint. The objective function goes like 10*x1^4+8*x2^5+12*x3^3+16*x4^6 Constraint is x1+x2+x3+x4<=100 For solving this I have introduced 8 auxiliary variables . 4 variables to equate with the main variables as constraints. Other 4 are used to raise the power of the first 4 constraints. But I am not getting the actual results. Kindly help me with this.

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• 

  Jaromił Najman

  Gurobi Staff

  • February 04, 2022 08:15

  Could you please share a minimal reproducible example where you model the objective function and don't get the correct solution?

  Best regards, 
  Jaromił

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  BISWAJIT KAR

  • February 06, 2022 08:05

  

  

  

  Hi Jaromił Najman, I have mentioned one objective and one constraint. I am sharing the code now

  m=gp.Model('Matelco')
  
  x1=m.addVar(vtype=GRB.CONTINUOUS, name ='x1')
  x2=m.addVar(vtype=GRB.CONTINUOUS, name ='x2')
  x3=m.addVar(vtype=GRB.CONTINUOUS, name ='x3')
  x4=m.addVar(vtype=GRB.CONTINUOUS, name ='x4')
  
  a=m.addVar(vtype= GRB.CONTINUOUS, name='a')
  b=m.addVar(vtype= GRB.CONTINUOUS, name='b')
  c=m.addVar(vtype= GRB.CONTINUOUS, name='c')
  d=m.addVar(vtype= GRB.CONTINUOUS, name='d')
  
  m.addConstr(a==x1, name =='c1')
  m.addConstr(b==x2, name =='c2')
  m.addConstr(c==x3, name =='c3')
  m.addConstr(d==x4, name =='c4')
  
  m.addGenConstrPow(x1, a, 4)
  m.addGenConstrPow(x2, b, 5)
  m.addGenConstrPow(x3, c, 3)
  m.addGenConstrPow(x4, d, 6)
  
  m.addConstr(x1+x2+x3+x4<=100, name = 'c1')
  m.setObjective(a+b+c+d, GRB.MAXIMIZE)
  
  m.update()
  m.optimize()
  m.printAttr('X')

  I am getting different results. The actual results are:

  x1=0
  x2=0
  x3=0
  x4=100

  But now I am getting different.

   

  Your correction would be highly appreciated.

  Thank You

  Regards,

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  Jaromił Najman

  Gurobi Staff

  • February 07, 2022 09:13
  • Edited

  Hi Biswajit,

  You could use the write method to generate an LP file and analyze your model in detail. In the LP file, you can see that what you actually modeled is

  \[\begin{align}
  \max \;&a + b + c +d\\
  x_1 &=a\\
  x_2 &=b\\
  x_3 &=c\\
  x_4 &=d\\
  a &= x_1^4\\
  b &= x_2^5\\
  c &= x_3^3\\
  d &= x_4^6\\
  100 &\geq x_1 + x_2 + x_3 + x_4\\
  x_i &\geq 0, a,b,c,d \geq 0
  \end{align}\]

  There you can see that the equality constraints

  \[\begin{align}
  x_1 &=a\\
  x_2 &=b\\
  x_3 &=c\\
  x_4 &=d\\
  \end{align}\]

  are wrong as they don't belong to the model you are trying to model. Additionally, the coefficients in your objective function are missing.

  import gurobipy as gp
  from gurobipy import GRB
  
  m=gp.Model('Matelco')
  m.setParam("FuncMaxVal",1e12)
  
  x1=m.addVar(vtype=GRB.CONTINUOUS, name ='x1')
  x2=m.addVar(vtype=GRB.CONTINUOUS, name ='x2')
  x3=m.addVar(vtype=GRB.CONTINUOUS, name ='x3')
  x4=m.addVar(vtype=GRB.CONTINUOUS, name ='x4')
  
  a=m.addVar(vtype=GRB.CONTINUOUS, name='a')
  b=m.addVar(vtype=GRB.CONTINUOUS, name='b')
  c=m.addVar(vtype=GRB.CONTINUOUS, name='c')
  d=m.addVar(vtype=GRB.CONTINUOUS, name='d')
  
  m.addGenConstrPow(x1, a, 4)
  m.addGenConstrPow(x2, b, 5)
  m.addGenConstrPow(x3, c, 3)
  m.addGenConstrPow(x4, d, 6)
  
  m.addConstr(x1+x2+x3+x4<=100, name = 'c5')
  m.setObjective(10*a + 8*b + 12*c + 16*d, GRB.MAXIMIZE)
  
  m.update()
  m.optimize()
  m.write("myLP.lp")
  m.printAttr('X')

  Note that you have to set the parameter FuncMaxVal to the maximum value that can be attained by your general constraint, which in your case equals \(100^6 = 10^12\).

  Working with such large values is not recommended as it may lead to numerical difficulties. You should try to reformulate or re-scale your model. Please refer to Guidelines for Numerical Issues for more details. Could you, use smaller values for powers, e.g., by finding a lower degree polynomial with same properties as the one you are currently using?

  Best regards, 
  Jaromił

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  BISWAJIT KAR

  • February 07, 2022 11:40

  

  

  

  Hi Jaromił Najman, I am very grateful to you for the clarification that you have provided.  I am afraid that the question is an assignment problem and it cannot be changed. But I want to thank you for the clarification. I would try to solve smaller questions.

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