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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

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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.scatter(x, y, results)


[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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