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# Xin-She Yang N. 4 Function

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 global minimum f(x*)­ =−1 are located at x*=(0,.., 0).

Characteristics

• The function is not convex.

• The function is defined on n-dimensional space.

• The function is non-differentiable.

• The function is non-separable.

Python Implementation

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#For n=2
#xinSheYangN4 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 xinSheYangN4(x1,x2):
return   (np.sin(x1)*np.sin(x1)-np.exp(-(x1**2+x2**2)))*np.exp(-((pow(np.sin(np.sqrt(abs(x1))),2))))+(np.sin(x2)*np.sin(x2)-np.exp(-(x1**2+x2**2)))*np.exp(-pow(np.sin(np.sqrt(abs(x2))),2))

x1=np.linspace(-10,10,30)
x2=np.linspace(-10,10,30)

X1,X2=np.meshgrid(x1,x2)
Z=xinSheYangN4(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('Xin-She Yang 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:

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

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