Mathematical Definition
Input Domain
It can be defined into any input domain but usually perm 0, d, beta function evaluated on the hypercube 𝑥𝑖 𝜖 [−𝑑, 𝑑], for all 𝑖 = 1, … . , 𝑑.
Global Minima
The function has one global minimum 𝑓(𝑥 ∗) = 0, 𝑎𝑡 𝑥 ∗ = (1, 1 /2 , … , 1 /𝑑 )
Description and Features
The function is defined on 2- dimensional space
Python Implementation
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% Author: Ekta Kumari
import matplotlib.pyplot as plt
import numpy as np
from numpy import sin
from numpy import e
from numpy import meshgrid
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
def f(x1,x2):
return (11*(x1-1)+12*(x2-0.5))**2 + (11*(x1**2-1)+12*(x2**2-0.25))**2
x1=np.linspace(-2,2)
x2=np.linspace(-2,2)
r_min,r_max=1,2
x1,x2=np.meshgrid(x1,x2)
results=f(x1,x2)
figure=plt.figure(figsize=(9,9))
axis=figure.gca(projection='3d')
axis.contour3D(x1, x2, results,450)
axis.set_title('Perm function 0,d,beta')
axis.view_init(40,40)
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.