Mathematical Definition
Input Domain
It can be defined into any input domain but usually evaluated on the hypercube 𝑥𝑖 ∈ [−65.536,65.536], for all 𝑖 = 1, … . , 𝑑.
Global Minima
The function has one global minimum 𝑓(𝑥 ∗ ) = 0, 𝑎𝑡 𝑥 ∗ = (0, … ,0).
Description and Features
The function is continuous function.
The function is convex function.
The function is referred as sum square’s function.
It is also an extension of the axis parallel hyper- ellipsoid function.
The function is unimodal function.
Python Implementation
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import matplotlib.pyplot as plt
import numpy as np
from numpy import *
from numpy import cos
from numpy import pi
from numpy import abs,sqrt
from numpy import meshgrid
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
def f(x1,x2):
return (2*(x1*x1)+(x2*x2))
x1= np.linspace(-65.536,65.536,500)
x2= np.linspace(-65.536,65.536,500)
r_min,r_max=-65.536,65.536
x1,x2=np.meshgrid(x1,x2)
results=f(x1,x2)
figure=plt.figure(figsize=(9,9))
axis=figure.gca(projection='3d')
axis.set_title('Rotated Hyper-Ellipsoid Function')
axis.contour3D(x1, x2, results,450)
axis.plot_surface(x1,x2,results, cmap=cm.YlGn)
axis.view_init(21,42)
axis.set_xlabel('X')
axis.set_ylabel('Y')
axis.set_zlabel('Z')
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.
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