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# Python Implementation of Shubert N. 4 Function

Updated: Jul 18, 2021   Mathematical Definition Input Domain

The function can be defined on any input range but it is usually evaluated on xi ∈ [-10,10] for

i = 1,...,n.

Global Minima

The function has one global minimum f(x*) ≈−25.740858.

Characteristics

The Shubert function has several local and global minima.

The function is continuous.

The function is not convex.

The function is defined on n-dimensional space.

The function is multimodal.

The function is differentiable.

The function is separable.

Python Implementation

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#For n=2
#shubertn4 accepts the   values of 2 MxM dimension matrices X1, X2
#it returns the   computation of the matrices in an MxM matrix Z
#the function is then   plotted using (X1,X2,Z)
#thus giving us a   contour plot

from mpl_toolkits import mplot3d
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm

def shubertn4(x1,x2):
y=0;
for j in range(1,6):
y=y+j*np.cos((j+1)*x1+j)+   j*np.cos((j+1)*x2+j)
return y

x1=np.linspace(-10,10,20)
x2=np.linspace(-10,10,20)

X1,X2=np.meshgrid(x1,x2)
Z=shubertn4(X1,X2)

def plotFunction(e,a):
fig=plt.figure(figsize = [12,8])
ax=plt.axes(projection='3d')
surf=ax.plot_surface(X1,X2,Z,cmap=cm.coolwarm)
ax.view_init(elev=e,azim=a)
ax.set_xlabel('X1')
ax.set_ylabel('X2')
ax.set_zlabel('fx')
ax.set_title('Shubert N. 4 Function')
fig.colorbar(surf, shrink=0.5, aspect=5)
plt.show()
plt.contour(X1, X2, Z)
plt.show()

from ipywidgets import interactive
iplot=interactive(plotFunction,
e=(-90,90,5),
a=(-90,90,5))
iplot

```

References:

 Survajonic, Sonja & Bingham, Derek, “Virtual Library of Simulation Experiments”, sfu.ca,

https://www.sfu.ca/~ssurjano/optimization.html