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Perform optimization using SciPy's optimize module to find function minimum

NumPy: Integration with SciPy Exercise-14 with Solution

Write a NumPy program to generate a set of points and perform optimization using SciPy's optimize module to find the minimum of a function.

Sample Solution:

Python Code:

import numpy as np  # Import NumPy library
from scipy.optimize import minimize  # Import minimize function from SciPy's optimize module

# Define the function to be minimized
def objective_function(x):
    return x[0]**2 + x[1]**2 + 3 * np.sin(x[0]) * np.cos(x[1])

# Generate a set of initial points
initial_guess = np.array([1.0, 1.0])

# Perform optimization using SciPy's minimize function
result = minimize(objective_function, initial_guess, method='BFGS')

# Print the optimization results
print("Optimization Result:")
print("Minimum Value of the Function:", result.fun)
print("Point at Minimum:", result.x)
print("Number of Iterations Performed:", result.nit)
print("Convergence Status:", result.success)
print("Optimization Message:", result.message)

Output:

Optimization Result:
Minimum Value of the Function: -1.5404628057094898
Point at Minimum: [-9.14856375e-01 -6.01451321e-08]
Number of Iterations Performed: 8
Convergence Status: True
Optimization Message: Optimization terminated successfully.

Explanation:

  • Import libraries:
    • Import the NumPy library for numerical operations.
    • Import the minimize function from SciPy's optimize module for performing optimization.
  • Define the Objective Function:
    • Create the function objective_function(x) to be minimized. This function takes a point x (array of coordinates) as input and returns the function value.
  • Generate Initial Points:
    • Define an initial guess as a NumPy array [1.0, 1.0] for the starting point of the optimization algorithm.
  • Perform optimization:
    • Use the minimize function from SciPy to find the minimum of the objective function starting from the initial guess. The method parameter is set to 'BFGS', a popular optimization algorithm.
  • Finally print the results of the optimization, including:
    • The minimum value of the function (result.fun).
    • The point at which the minimum occurs (result.x).
    • The number of iterations performed (result.nit).
    • The convergence status (result.success).
    • The optimization message (result.message).

Python-Numpy Code Editor: