Mathematical Definition
Input Domain
The function is defined on input range xi ∈[−6, 6] for i=1, 2.
Global Minima
The Himmelblau Function has four identical local minimum at:
f(x∗)=0 at x∗=(3, 2)
f(x∗)=0 at x∗=(−2.805118, 3.283186)
f(x∗)=0 at x∗=(−3.779310, −3.283186)
f(x∗)=0 at x∗=(3.584458, −1.848126)
The function has one local maximum at x=-0.270845 and y=-0.923039 where f(x, y)=181.617.
Description and Features
This function is named after David Mautner Himmelblau who introduced it. The Himmelblau Function is defined on the two dimensional space, used to test the performance of optimization algorithms such as:
Convergence rate
Precision
Robustness
The Himmelblau function is a
Multi-modal function
Non-separable
Non-convex
Continuous
Python Implementation
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% Author: RIYA SHIVHARE
import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits import mplot3d
def f(x,y):
return (((x**2+y-11)**2) + (((x+y**2-7)**2)))
X=np.linspace(-6,6)
Y=np.linspace(-6,6)
x,y=np.meshgrid(X,Y)
F=f(x,y)
fig =plt.figure(figsize=(9,9))
ax=plt.axes(projection='3d')
ax.contour3D(x,y,F,450)
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('F')
ax.set_title('Himmelblau Function')
ax.view_init(50,50)
#plt.contour(x,y,F,15)
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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