Why Gurobi cannot find the solution to a simple Quadratic Optimization Problem?

Answered

Zhongjin Lu

• September 11, 2024 01:01

import gurobipy as gp
from gurobipy import GRB
import numpy as np
import scipy.sparse as sp

# Create a new model
m = gp.Model("matrix1")

# Create variables
x = m.addMVar(shape=7, vtype=GRB.CONTINUOUS, name="x")
# Set objective
obj = np.array([1, -0.1, 0.8, 0.3, 0.9,0.4,0.4])*0.01 # expected monthly returns
m.setObjective(obj @ x , GRB.MAXIMIZE) #50bps per trade; 50bps per month for shorting

Sigma_psd = np.diag(np.array([30, 0.2, 20, 16, 32, 28, 19])*0.0001)
# Define the quadratic constraint; 5% per month for the portfolio
V = (0.05) ** 2

# Add the quadratic constraint to the model
m.addConstr(x @ Sigma_psd @ x == V, name="quadratic_constraint")


# Optimize model
m.optimize()
weights = x.X
print("Weights from Gurobi:", ", ".join(f"{w:g}" for w in weights))
print(f"Obj: {m.ObjVal:g}")


Sigma_inv = np.linalg.inv(Sigma_psd)  # Inverse of Sigma
weights = np.dot(Sigma_inv, obj)   # Dot product of inv(Sigma) and obj
weights = np.sqrt(V/np.dot(weights.T, np.dot(Sigma_psd, weights)))*weights
print("Correct Weights:", ", ".join(f"{w:g}" for w in weights))
print(f"Obj under Correct Weights: {np.dot(weights.T, obj):g}")
 

Related to

• QCP

0

• 

  Ahmed Azab

  Gurobi Staff

  • September 11, 2024 02:01
  • Edited

  Hi,
  I solved your model using Gurobi version 11.0.3. 

  Optimal solution found (tolerance 1.00e-04)
  Best objective 1.661528493733e-02, best bound 1.661528493733e-02, gap 0.0000%

  Note that the default lower bound value for the variables is 0.
  
  Regards
  -Ahmed
  
  
  

  0
• 

  Zhongjin Lu

  • September 11, 2024 15:07

  

  

  Ahmed,

  Thanks a lot. I didn't realize that the default lower bound value is 0. Once I relax it, Gurobi finds the correct answer. Thanks!

  Zhongjin

  0

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