Mathematical Definition
Input Domain
The function can be defined on any input domain but it is usually evaluated on xi∈[−500,500] for i=1,…,n. Here, n = 2.
Global Minima
The global minima f(x*)=0 are located at x*=(±√i,…,±√i)
Description and Features
The function is continuous, not convex, defined on n-dimensional space, multimodal, differentiable, non-separable.
Python Implementation
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import numpy as np
import matplotlib.pyplot as plt
fig = plt.figure()
ax = fig.gca(projection='3d')
x = np.arange(-500, 500, 0.1)
y = np.arange(-500, 500, 0.1)
x, y = np.meshgrid(x, y)
res = 0
for i in range(3):
res = res + ((x ** 2 + y ** 2) -i) ** 2
ax.plot_surface(x, y, res, cmap='jet')
plt.show()
plt.contour(x,y,res)
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.