Mathematical Definition
Input Domain
It can be defined on any input domain but it’s usually evaluated on the square 𝑥𝑖 ∈ [−100,100] for i=1,2
Global Minima
It has one local minima at 𝑓(𝑥 ∗ ) = 0 𝑎𝑡 𝑥 ∗ = (0,0).
Description and Features
Bohachevsky functions are continuous.
The function is defined on 2- dimensional space.
Bohachevsky functions are unimodal.
The functions all have the same similar bowl shape
Python Implementation
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% Author: Ekta Kumari
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
def f( x1,x2):
return x1**2 +2*(x2**2)-0.3*cos(3*pi*x1)-0.4*cos(4*pi*x2)+0.7
x1=np.linspace(-100,100,500)
x2=np.linspace(-100,100,500)
r_min,r_max=-100,100
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('Bohachevsky function')
axis.view_init(21,40)
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.
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