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Rotated Hyper-Ellipsoid Function

Updated: Mar 30, 2022



Mathematical Definition


Input Domain


It can be defined into any input domain but usually evaluated on the hypercube ๐‘ฅ๐‘– โˆˆ [โˆ’65.536,65.536], for all ๐‘– = 1, โ€ฆ . , ๐‘‘.


Global Minima


The function has one global minimum ๐‘“(๐‘ฅ โˆ— ) = 0, ๐‘Ž๐‘ก ๐‘ฅ โˆ— = (0, โ€ฆ ,0).


Description and Features


The function is continuous function.

The function is convex function.

The function is referred as sum squareโ€™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.




#optimization #benchmarkfunction

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