Mathematical Definition

\[f(\mathbf{x})=f(x_1, ...,x_n)=\prod_{i=1}^{n}{\left(\sum_{j=1}^5{ cos((j+1)x_i+j)}\right)}\]

Plots

Two contours of the function are presented below:

Description and Features

• The function is continuous.
• The function is not convex.
• The function is defined on n-dimensional space.
• The function is multimodal.
• The function is differentiable.
• The function is non-separable.

Input Domain

The function can be defined on any input domain but it is usually evaluated on $x_i \in [-10, 10]$ for $i=1, …, n$.

Global Minima

The function has 18 global minima $f(\textbf{x}^{\ast})\approx-186.7309$.

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 shubert

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

MATLAB

An implementation of the Shubert Function with MATLAB is provided below.

% Computes the value of the Shubert function.
% SCORES = SHUBEERTFCN(X) computes the value of the Shubert 
% function at point X. SHUBEERTFCN accepts a matrix of size M-by-N 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 = shubertfcn(x)
    n = size(x, 2);
    
    scores = 1;
    for i = 1:n
        inner_sum = 0;
        for j = 1:5
            inner_sum = inner_sum + j * cos(((j + 1) * x(:, i)) + j);
        end
        scores = inner_sum .* scores;
    end
end

The function can be represented in Latex as follows:

f(\mathbf{x})=f(x_1, ...,x_n)=\prod_{i=1}^{n}{\left(\sum_{j=1}^5{ cos((j+1)x_i+j)}\right)}

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
• E. P. Adorio, U. P. Dilman, “MVF - Multivariate Test Function Library in C for Unconstrained Global Optimization Methods,” [Available Online]: http://www.geocities.ws/eadorio/mvf.pdf