ACKLEY N.3 FUNCTION

Mathematical Definition

Input Domain

The function is defined on input domain i.e. x ∈ [−32, 32] and y ∈ [−32, 32].

Global Minima

The function has two global minima at f(z) = −195.629028238419 located at z =

(±0.682584587365898, -0.36075325513719).

Characteristics

The function is convex.

The function is differentiable.

The function is non-separable.

The function is defined on 2-dimensional space.

Python Implementation

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from mpl_toolkits import mplot3d
import matplotlib.pyplot as plt
import numpy as np
from matplotlib import cm

def f(x,y):
    return -200*np.exp(-0.2*np.sqrt(x**2 + y**2)) + 
    5*np.exp(np.cos(3*x)+np.sin(3*y))

X = np.linspace(-32,32)
Y = np.linspace(-32,32)
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, cmap=cm.flag_r)
ax.set_title('Ackley N.3 Function')
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('F')
ax.view_init(21,45)
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.

#optimization #benchmarkfunction