Mathematical Definition

\[f(\textbf{x}) = f(x_1, x_2, ..., x_n) = 418.9829d -{\sum_{i=1}^{n} x_i sin(\sqrt{|x_i|})}.\]

Plots

The contour of the function:

Description and Features

• The function is continuous.
• The function is not convex.
• The function can be defined on n-dimensional space.
• The function is multimodal.
• The function is not .

Input Domain

The function can be defined on any input domain but it is usually evaluated on the hypercube $x_i \in [-500, 500]$ for $i = 1..n$.

Global Minima

$f(\textbf{x}^{\ast}) = 0$ at $\textbf{x}^{\ast} = (420.9687, …, 420.9687)$

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 schwefel

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

MATLAB

An implementation of the Schwefel Function with MATLAB is provided below. Schwefel Function can be implemented with a for loop that iterates over all the components of the input vector but, MATLAB and Octave have built-in facilities that makes the implementation more efficient and concise.

% Computes the value of the Schwefel benchmark function.
% SCORES = SCHWEFELFCN(X) computes the value of the Schwefel function at 
% point X. SCHWEFELFCN 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
function scores = schwefelfcn(x)
    n = size(x, 2);
    scores = 418.9829 * n - (sum(x .* sin(sqrt(abs(x))), 2));
end

The function can be represented in Latex as follows:

f(\textbf{x}) = f(x_1, x_2, ..., x_n) = 418.9829d -{\sum_{i=1}^{n} x_i sin(\sqrt{|x_i|})}

References:

• http://www.sfu.ca/~ssurjano