Mathematical Definition
Here m and β are the function parameters and usually taken as m=5 and β= 15.
Input Domain
Defined within the domain -2π < xi < 2π, for i = 1,2, ..., n. It can be defined on any input range as well but preferred to be as given above.
Global Minima
It has many local minima and the unique global minimum f(x*) = -1 at x* = (0,0,..., 0) for m=5 and β = 15.
Description and Features
Uni-model function.
Differentiable.
Convex.
Non-separable
Smooth
Python Implementation
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% Author: Vanshita Tripathi
import matplotlib.pyplot as plt
import matplotlib as mpl
from mpl_toolkits import mplot3d
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
from numpy import*
%matplotlib notebook
plt.rcParams['figure.figsize'] = (6,4)
plt.rcParams['figure.dpi']=150
fig=plt.figure()
ax=fig.add_subplot(111,projection='3d')
ax= plt.axes(projection='3d')
def f(x1,x2):
a = exp( -(x1/15)**10)-2 * exp ( (-x1**2) (x2**2)) * cos(x1) * cos(x1) * cos(x2) * cos(x2)
return a
x1= linspace(-2*pi,2*pi)
x2= linspace(-2*pi,2*pi)
X1,X2= meshgrid(x1,x2)
ax.plot_surface(X1,X2,f(X1,X2), cmap='jet')
ax.set_xlabel('x1')
ax.set_ylabel('x2')
ax.set_zlabel('f(x1,x2)')
ax.view_init(10,10)
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.