Mathematical Definition

\[f(x, y) = - \frac{1 + cos(12\sqrt{x^{2} + y^{2}})}{(0.5(x^{2} + y^{2}) + 2)}\]

Plots

The contour of the function:

Description and Features

• The function is continuous.
• The function is not convex.
• The function is defined on 2-dimensional space.
• The function is multimodal.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [-5.2, 5.2]$ for $i = 1, 2$.

Global Minima

$f(\textbf{x}^{\ast}) = -1$ at $\textbf{x}^{\ast} = (0, 0)$

Implementation

Python

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

from benchmarkfcns import dropwave

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

MATLAB

An implementation of the Drop-Wave Function with MATLAB is provided below.

% Computes the value of the Drop-Wave benchmark function.
% SCORES = DROPWAVEFCN(X) computes the value of the Drop-Wave function at 
% point X. DROPWAVEFCN 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.
% For more information please visit: 
% 
% 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 = dropwavefcn(x)
    n = size(x, 2);
    assert(n == 2, 'Drop-Wave function is only defined on a 2D space.')
    X = x(:, 1);
    Y = x(:, 2);
    
    numeratorcomp = 1 + cos(12 * sqrt(X .^ 2 + Y .^ 2));
    denumeratorcom = (0.5 * (X .^ 2 + Y .^ 2)) + 2;
    scores = - numeratorcomp ./ denumeratorcom;
end

The function can be represented in Latex as follows:

f(x, y) = - \frac{1 + cos(12\sqrt{x^{2} + y^{2}})}{(0.5(x^{2} + y^{2}) + 2)}

References:

• http://www.sfu.ca/~ssurjano