Mathematical Definition

\[f(x, y) = -200e^{-0.2\sqrt{x^2 + y^2}}\]

Plots

Ackley N. 2 Function

A contour of the function is presented below:

Ackley N. 2 Function

Description and Features

The function is convex.
The function is defined on 2-dimensional space.
The function is non-separable.
The function is differentiable.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [-32, 32]$ for $i=1, 2$.

Global Minima

The function has a global minimum at $f(\textbf{x}^{\ast})=-200$ located at $\mathbf{x^\ast}=(0, 0)$.

Implementation

Python

For Python, the function is implemented in the benchmarkfcns package and can be installed from command line with pip install benchmarkfcns.

from benchmarkfcns import ackley2

print(ackley2([[0, 0],
              [1, 1]]))

MATLAB

An implementation of the Ackley N. 2 Function with MATLAB is provided below.

% Computes the value of the Ackley N. 2 function.
% SCORES = ACKLEYN2FCN(X) computes the value of the Ackley N. 2
% function at point X. ACKLEYN2FCN accepts a matrix of size M-by-2 and 
% returns a vetor SCORES of size M-by-1 in which each row contains the 
% function value for the corresponding row of X.
% 
% Author: Mazhar Ansari Ardeh
% Please forward any comments or bug reports to mazhar.ansari.ardeh at
% Google's e-mail service or feel free to kindly modify the repository.
function scores = ackleyn2fcn(x)
    
    n = size(x, 2);
    assert(n == 2, 'Ackley N. 2 function is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    scores = -200 * exp(-0.02 * sqrt((X .^ 2) + (Y .^ 2)));
end

The function can be represented in Latex as follows:

f(x, y) = -200e^{-0.2\sqrt{x^2 + y^2}}

References:

Momin Jamil and Xin-She Yang, A literature survey of benchmark functions for global optimization problems, Int. Journal of Mathematical Modelling and Numerical Optimisation}, Vol. 4, No. 2, pp. 150–194 (2013), arXiv:1308.4008
D. H. Ackley, “A Connectionist Machine for Genetic Hill-Climbing,” Kluwer, 1987.