Mathematical Definition

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

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 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 one global minimum $f(\textbf{x}^{\ast})\approx-25.740858$.

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 shubert4

print(shubert4([[0, 0, 0],
              [1, 1, 1]]))

MATLAB

An implementation of the Shubert 4 Function with MATLAB is provided below.

% Computes the value of the Shubert 4 function.
% SCORES = SHUBEERT4FCN(X) computes the value of the Shubert 4
% function at point X. SHUBEERT4FCN 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 = shubert4fcn(x)
    n = size(x, 2);
    
    scores = 0;
    for i = 1:n
        for j = 1:5
            scores = scores + j * cos(((j + 1) * x(:, i)) + j);
        end
    end
end

The function can be represented in Latex as follows:

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

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