# Global optimization in Python with scipy.optimize

Einblick Content Team - March 1st, 2023

In data analysis, finding the global minimum of a function is a common task. However, it can be challenging to find the optimal solution due to the presence of multiple local minima. The scipy library provides a convenient method for global optimization called scipy.optimize.basinhopping(). This function implements the basin-hopping algorithm, which uses a combination of local optimization methods and stochastic jumps to find the global minimum of a given function.

In this tutorial, we provide an example of using the scipy.optimize.basinhopping() function to find the global minimum of a one-dimensional multimodal function. The entire code can be found in the canvas below:

## 1. Import libraries

# Import the necessary libraries
import numpy as np
import scipy.optimize as opt

## 2. Define objective function and initial guess

We converted the following function into Python, and gave an initial guess of x = -2 for the location of the global minimum.

$f(x) = -10\cos(\pi x-2.2)+(x+1.5)x$
# Define objective function
def f(x):
return -10*np.cos(np.pi*x - 2.2) + (x + 1.5)*x

# Set initial guess
x0 = [-2]

## 3. Setup and call basinhopping() function

In this case, we call on the BFGS (Broyden, Fletcher, Goldfarb, and Shanno) method for unconstrained minimization. scipy implements a number of other methods as well.

We have also specified how many basin-hopping iterations to run via the argument niter.

# Set up args for basinhopping and call function
minimizer_kwargs = {"method": "BFGS"}
optimization_algorithm = opt.basinhopping(f, x0, minimizer_kwargs = minimizer_kwargs, niter = 200)

## 4. Print results

print("1-D function")
print(optimization_algorithm.message)

# Save results
optimized_x = optimization_algorithm.x
optimized_fun = optimization_algorithm.fun

# Print results
print("Optimized x: ", optimized_x)
print("Optimized function value: ", optimized_fun)

Output:

1-D function
requested number of basinhopping iterations completed successfully
Optimized x:  [-1.28879778]
Optimized function value:  -10.26631244852453

From the results, stored in the message, x, and fun attributes, we can see that the algorithm detected the global minimum at:

$x = -1.299 \newline f(x) = -10.266$

## BONUS. Plot function, and check results

To check that our function has not gotten caught at a local minimum, we can plot the function using the following code:

import seaborn as sns
import matplotlib.pyplot as plt
sns.set_theme()

# Generate data for objective function graph
X = np.arange(-10, 10, 0.2)
y = [f(x) for x in X]

# Plot global minimum
plt.vlines(x = optimized_x, ymin = -10, ymax = 125, colors = 'red')
plt.hlines(y = optimized_fun, xmin = -10, xmax = 10, colors = 'red')
plt.plot(X, y)
plt.plot(optimized_x, optimized_fun,'o', color = "black")

Output:

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