Shubert Function

Mathematical Definition

Input Domain

It can define into any input domain but usually its evaluated on the square 𝑥𝑖 ∈ [−10,10] for all i= 1,2

Global Minima

It has 18 global minima 𝑓(𝑥 ∗ ) ≈ −186.7309.

Description and Features

Shubert function is continuous function.

The function is differentiable.

The function is non-separable.

The function is defined on n – dimensional space.

Python Implementation

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import matplotlib.pyplot as plt
import numpy as np
from numpy import sin
from numpy import *
from numpy import pi
from numpy import sqrt
from matplotlib import cm

def f(x1,x2):
      sum1=0
      sum2=0
      for i in range(1,6):
          sum1 = sum1 + (i* cos(((i+1)*x1) +i))
          sum2 = sum2 + (i* cos(((i+1)*x2) +i))
      return sum1 * sum2
 
x1 =np.linspace(-10,10,100)
x2 =np.linspace(-10,10,100)
r_min,r_max= -10,10

x1,x2=np.meshgrid(x1,x2)
results=f(x1,x2)

figure=plt.figure(figsize=(9,9))
axis=figure.gca(projection='3d')
axis.contour3D(x1, x2, results,15)
axis.set_title('Shubert function')
axis.plot_surface(x1,x2,results, cmap=cm.rainbow)


axis.view_init(elev=21,azim=42)
axis.set_xlabel('X')
axis.set_ylabel('Y')
axis.set_zlabel('Z')
plt.contour(x1, x2, results,15)
plt.show()

References:

[1] Jamil, Momin, and Xin-She Yang. "A literature survey of benchmark functions for global optimization problems." International Journal of Mathematical Modelling and Numerical Optimization 4.2 (2013): 150-194.

#optimization #benchmarkfunction