Mathematical Definition
Input Domain
The function is usually evaluated at xi ∈ [-32.768, 32.768], for all i = 1, …, d, although it may also be restricted to a smaller domain. Here, d=2.
Global Minima
The global minimum of the function is at f(x* ) = 0, at x* = (0,,,,,,,,,,0)
Description and Features
The Ackley function is widely used for testing optimization algorithms. In its two-dimensional form, as shown in the plot above, it is characterized by a nearly flat outer region, and a large hole at the centre. The function poses a risk for optimization algorithms, particularly hill climbing algorithms, to be trapped in one of its many local minima. a = 20, b = 0.2 and c = 2π.
Python Implementation
% Please forward any comments or bug reports in chat
Copyright 2021. INDUSMIC PRIVATE LIMITED.THERE IS NO WARRANTY, EXPRESS OR IMPLIED. WE DO NOT ASSUME ANY LIABILITY FOR THE USE OF THIS PROGRAM. If software is modified to produce derivative works, such modified software should be clearly marked. Additionally, user can redistribute it and/or modify it under the terms of the GNU General Public License. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY. See the GNU General Public License for more details.
% for any support connect with us on help.indusmic@gmail.com
% Author: Sakshi Chadda
from numpy import arange
from numpy import exp
from numpy import sqrt
from numpy import cos
from numpy import e
from numpy import pi
from numpy import meshgrid
import matplotlib.pyplot as plt
def objective(x, y):
return -20.0 * exp(-0.2 * sqrt(0.5 * (x**2 + y**2)))-exp(0.5 * (cos(2 *
pi * x)+cos(2 * pi * y))) + e + 20
r_min, r_max = -32.768, 32.768
xaxis = arange(r_min, r_max, 2.0)
yaxis = arange(r_min, r_max, 2.0)
x, y = meshgrid(xaxis, yaxis)
results = objective(x, y)
figure = plt.figure()
axis = figure.gca( projection='3d')
axis.plot_surface(x, y, results, cmap='jet', shade= "false")
plt.show()
plt.contour(x,y,results)
plt.show()
plt.scatter(x, y, results)
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.