Mathematical Definition

\[f(\textbf{x}) = \sum_{i=1}^{n-1}(x_i^2)^{(x_{i+1}^{2}+1)}+(x_{i+1}^2)^{(x_{i}^{2}+1)}\]

Plots

Two contours of the function are presented below:

Description and Features

• The function is convex.
• The function is defined on n-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 $x_i \in [-1, 4]$ for $i=1, …, n$.

Global Minima

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

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 brown

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

MATLAB

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

% Computes the value of the Brown benchmark function.
% SCORES = BROWNFCN(X) computes the value of the Brown function at point X.
% BROWNFCN 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 = brownfcn(x)
    
    n = size(x, 2);  
    scores = 0;
    
    x = x .^ 2;
    for i = 1:(n-1)
        scores = scores + x(:, i) .^ (x(:, i+1) + 1) + x(:, i+1).^(x(:, i) + 1);
    end
end

The function can be represented in Latex as follows:

f(\textbf{x}) = \sum_{i=1}^{n-1}(x_i^2)^{(x_{i+1}^{2}+1)}+(x_{i+1}^2)^{(x_{i}^{2}+1)}

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