Differential Evolution Optimization Example in Python

    Optimization algorithms play an important role in various fields, including machine learning, engineering, and finance. Differential Evolution (DE) is a powerful optimization algorithm designed for global optimization of real-valued functions. 

    In this blog post, we'll explore the basics of Differential Evolution and demonstrate its application on a specific function using the SciPy differential_evolution() function in Python.

    The tutorial covers:

1. Understanding the problem
2. Differential Evolution implementation
3. Conclusion
   
4. Source code listing
   

 We'll start by loading the required libraries.

import numpy as np
from scipy.optimize import differential_evolution
import matplotlib.pyplot as plt
from matplotlib import cm

 

Understanding the problem

 

   We'll start creating the objective function to optimize and visualize it in a 3D plot.  In below code, we'll define ranges and function in a mesh grid then visualize it in a plot.

 

# define ranges
x_range = np.arange(-4, 4, 0.1)
y_range = np.arange(-4, 4, 0.1)

# create meshgrid
x, y = np.meshgrid(x_range, y_range)

# define the function 
z = np.sqrt(np.sqrt(x**2+y**2))

# Plot the surface.
fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
surf = ax.plot_surface(x, y, z, cmap=cm.jet,
                       linewidth=0, antialiased=False)

fig.colorbar(surf, shrink=0.5, aspect=5)
plt.show()

 

Our task is to search a global minimum of the above function in given range. 

 

Differential Evolution implementation

 

    Differential Evolution (DE) is a population-based metaheuristic search algorithm to find the global minimum of a multivariate function. DE is a kind of evolutionary computing algorithm that starts with an initial set of candidate solution and updates it iteratively. 

 

    SciPy provides the differential_evolution() function for implementing differential evolution, and we'll use it to find the minimum of a given function in this tutorial. The function requires search bounds and callable objective function. We'll define function without changing the formula above. 

 

 
# define function
def func(p):
    x, y = p
    r = np.sqrt(x**2+y**2)
    return np.sqrt(r)

We'll set variable bounds that can be specified by defining the max and min values.

bounds = [[-4, 4], [-4, 4]]

Next, execute the differential evolution function with the given objective function and bounds.

# execute differential evolution search
result = differential_evolution(func, bounds)
 

print(result)

     fun: 0.0
 message: 'Optimization terminated successfully.'
    nfev: 3093
     nit: 100
 success: True
       x: array([0., 0.]) 

The result shows that minimum of the function is located in a point of (0, 0). 

   Here, result contains the following attributes:

    fun - value of objective function 

    nfev -  the number of evaluation, 

    nit - the number of iterations, 

    success - the existence of optimizer, 

    x -  the solution of the optimization

To print evaluated function at every iteration, we'll set true to the 'disp' parameter.

result = differential_evolution(func, bounds, disp=True)

differential_evolution step 1: f(x)= 0.681272
differential_evolution step 2: f(x)= 0.457103
differential_evolution step 3: f(x)= 0.457103
differential_evolution step 4: f(x)= 0.301956
differential_evolution step 5: f(x)= 0.301956

...

differential_evolution step 83: f(x)= 5.01213e-08
differential_evolution step 84: f(x)= 3.54411e-08
differential_evolution step 85: f(x)= 2.98023e-08
differential_evolution step 86: f(x)= 0
differential_evolution step 87: f(x)= 0
differential_evolution step 88: f(x)= 0
differential_evolution step 89: f(x)= 0
differential_evolution step 90: f(x)= 0
differential_evolution step 91: f(x)= 0
differential_evolution step 92: f(x)= 0
differential_evolution step 93: f(x)= 0 

 

    Now we can visualize the detected minimum of a function in 3D scatter plot.
 

# Scatter plot the optimal point
optimal_point = result.x
z_value = func(optimal_point)

ax.scatter(optimal_point[0], optimal_point[1], z_value, 

            color='red', marker='o', s=70, label='Optimal Point')
fig.colorbar(surf, shrink=0.5, aspect=5)
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.legend()
plt.show() 

 

Conclusion

     In this tutorial, we've briefly learned how to use differential evolution method with differential_evolution() function in Python. Differential Evolution is a versatile optimization algorithm suitable for various optimization problems. The full source code is listed below. 

 

 

Source code listing

import numpy as np
from scipy.optimize import differential_evolution
import matplotlib.pyplot as plt
from matplotlib import cm

# define ranges
x_range = np.arange(-4, 4, 0.1)
y_range = np.arange(-4, 4, 0.1)

# create meshgrid
x, y = np.meshgrid(x_range, y_range)

# define the function
z = np.sqrt(np.sqrt(x**2 + y**2))

# plot the surface
fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
surf = ax.plot_surface(x, y, z, cmap=cm.jet, linewidth=0, antialiased=False)


# set bounds
bounds = [[-4, 4], [-4, 4]]

# define function
def func(p):
    x, y = p
    r = np.sqrt(x**2 + y**2)
    return r

# execute differential evolution search 
result = differential_evolution(func, bounds, disp=True)
print(result)

# Scatter plot the optimal point
optimal_point = result.x
z_value = func(optimal_point)

ax.scatter(optimal_point[0], optimal_point[1], z_value, color='red', 

         marker='o', s=70, label='Optimal Point')
fig.colorbar(surf, shrink=0.5, aspect=5)
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.legend()
plt.show()

 

 

References:

1. SciPy Differential Evolution