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# Xin-She Yang N.3 Function

Mathematical Definition

Here m and β are the function parameters and usually taken as m=5 and β= 15.

Input Domain

Defined within the domain -2π < xi < 2π, for i = 1,2, ..., n. It can be defined on any input range as well but preferred to be as given above.

Global Minima

It has many local minima and the unique global minimum f(x*) = -1 at x* = (0,0,..., 0) for m=5 and β = 15.

Description and Features

• Uni-model function.

• Differentiable.

• Convex.

• Non-separable

• Smooth

Python Implementation

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import matplotlib.pyplot as plt
import matplotlib as mpl
from mpl_toolkits import mplot3d
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
from numpy import*

%matplotlib notebook
plt.rcParams['figure.figsize'] = (6,4)
plt.rcParams['figure.dpi']=150
fig=plt.figure()

ax= plt.axes(projection='3d')

def f(x1,x2):
a = exp( -(x1/15)**10)-2 * exp ( (-x1**2) (x2**2)) * cos(x1) * cos(x1) * cos(x2) * cos(x2)
return a

x1= linspace(-2*pi,2*pi)
x2= linspace(-2*pi,2*pi)
X1,X2= meshgrid(x1,x2)
ax.plot_surface(X1,X2,f(X1,X2), cmap='jet')
ax.set_xlabel('x1')
ax.set_ylabel('x2')
ax.set_zlabel('f(x1,x2)')
ax.view_init(10,10)
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.