Mathematical Definition

\[f(\mathbf x)=f(x_1, ..., x_n)=\left(\sum_{i=1}^{n}sin^2(x_i)-e^{-\sum_{i=1}^{n}x_i^2}\right)e^{-\sum_{i=1}^{n}{sin^2\sqrt{|x_i|}}}\]

Plots

Two contours of the function are presented below:

Description and Features

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

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 global minimum $f(\textbf{x}^{\ast})=-1$ are located at $\mathbf{x^\ast}=(0, …, 0)$.

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 xinsheyang4

print(xinsheyang4([[0, 0, 0],
              [1, 1, 1]]))

MATLAB

An implementation of the Xin-She Yang N. 4 Function with MATLAB is provided below.

% Computes the value of the Xin-She Yang N. 4 function.
% SCORES = XINSHEYANGN4FCN(X) computes the value of the Xin-She Yang N. 4
% function at point X. XINSHEYANGN4FCN 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 = xinsheyangn4fcn(x)
     scores = (sum(sin(x) .^2, 2) - exp(-sum(x .^ 2, 2))) .* exp(-sum(sin(sqrt(abs(x))) .^2, 2));
end 

The function can be represented in Latex as follows:

f(\mathbf x)=f(x_1, ..., x_n)=\left(\sum_{i=1}^{n}sin^2(x_i)-e^{-\sum_{i=1}^{n}x_i^2}\right)e^{-\sum_{i=1}^{n}{sin^2\sqrt{|x_i|}}}

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
• X. S. Yang, “Test Problems in Optimization,” Engineering Optimization: An Introduction with Metaheuristic Applications John Wliey & Sons, 2010. [Available Online]: http://arxiv.org/abs/1008.0549