Xin-She Yang's Function No.01

I. Mathematical Expression:

$$f(X)=\left[e^{\displaystyle -\sum^n_{i=1}\left(\frac{x_i}{\beta}\right)^{2m}}-2e^{\displaystyle -\sum^n_{i=1}\left(x_i-c\right)^2}\right]\cdot \prod^n_{i=1}\cos^2\left(x_i\right)$$

where:

\(\bullet\) \(c=0, \pi\) or any other shifting value, \(m=5\) and \(\beta=15\)

\(\bullet\) \(-20\leq x_i \leq 20\) , \(i=1,2,\cdots,n\)

\(\bullet\) \(f_{min}(x^*)=-1\)

\(\bullet\) \(x^*_i=c\) \(\rightarrow\) In the literature, this \(c\) is mostly considered either equal to 0 (i.e., the origin) or \(\pi\)

\(\bullet\) This is the \(1^{st}\) Xin-She Yang's Function as can be seen from his first published journal paper in [1]. Thus, the sorting of the given functions in [3, 4] is not correct. Also, the expression given in [3, 4] is not correct, because the product is located outside the brackets [5].

II. Citation Policy:

If you publish material based on databases obtained from this repository, then, in your acknowledgments, please note the assistance you received by using this repository. This will help others to obtain the same data sets and replicate your experiments. We suggest the following pseudo-APA reference format for referring to this repository:

Ali R. Al-Roomi (2015). Unconstrained Single-Objective Benchmark Functions Repository [https://www.al-roomi.org/benchmarks/unconstrained]. Halifax, Nova Scotia, Canada: Dalhousie University, Electrical and Computer Engineering.

Here is a BiBTeX citation as well:

@MISC{Al-Roomi2015,
author = {Ali R. Al-Roomi},
title = {{Unconstrained Single-Objective Benchmark Functions Repository}},
year = {2015},
address = {Halifax, Nova Scotia, Canada},
institution = {Dalhousie University, Electrical and Computer Engineering},
url = {https://www.al-roomi.org/benchmarks/unconstrained}
}

III. 2&3D-Plots:

IV. Controllable 3D Model:

- In case you want to adjust the rendering mode, camera position, background color or/and 3D measurement tool, please check the following link

- In case you face any problem to run this model on your internet browser (it does not work on mobile phones), please check the following link

V. MATLAB M-File:

% Xin-She Yang's Function # 1
% Range of initial points: -20 <= xj <= 20 , j=1,2,...,n
% Global minima: (x1,x2,...,xn)=c
% f(X)=-1
% Coded by: Ali R. Alroomi | Last Update: 12 August 2015 | www.al-roomi.org
    
clear
clc
warning off
 
c=pi;
m=5;
beta=15;
 
x1min=-20;
x1max=20;
x2min=-20;
x2max=20;
R=1500; % steps resolution
x1=x1min:(x1max-x1min)/R:x1max;
x2=x2min:(x2max-x2min)/R:x2max; 
    
for j=1:length(x1)
    
    % For 1-dimensional plotting
    f1(j)=(exp(-(x1(j)/beta)^(2*m))-2*exp(-(x1(j)-c)^2))*(cos(x1(j)))^2;
    
    % For 2-dimensional plotting
    for i=1:length(x2)
        fn(i)=(exp(-(x1(j)/beta)^(2*m)-(x2(i)/beta)^(2*m))- ...
            2*exp(-(x1(j)-c)^2-(x2(i)-c)^2))*(cos(x1(j)))^2*(cos(x2(i)))^2;
    end
    
    fn_tot(j,:)=fn;
 
end
 
figure(1)
plot(x1,f1,'r','LineWidth',2);set(gca,'FontSize',12);
xlabel('x','FontName','Times','FontSize',20,'FontAngle','italic');
ylabel('f(x)','FontName','Times','FontSize',20,'FontAngle','italic');
title('2D View','FontName','Times','FontSize',24,'FontWeight','bold');
 
figure(2)
meshc(x1,x2,fn_tot);colorbar;set(gca,'FontSize',12);
xlabel('x_2','FontName','Times','FontSize',20,'FontAngle','italic');
set(get(gca,'xlabel'),'rotation',25,'VerticalAlignment','bottom');
ylabel('x_1','FontName','Times','FontSize',20,'FontAngle','italic');
set(get(gca,'ylabel'),'rotation',-25,'VerticalAlignment','bottom');
zlabel('f(X)','FontName','Times','FontSize',20,'FontAngle','italic');
title('3D View','FontName','Times','FontSize',24,'FontWeight','bold');
 
figure(3)
mesh(x1,x2,fn_tot);view(0,90);colorbar;set(gca,'FontSize',12);
xlabel('x_2','FontName','Times','FontSize',20,'FontAngle','italic');
ylabel('x_1','FontName','Times','FontSize',20,'FontAngle','italic');
zlabel('f(X)','FontName','Times','FontSize',20,'FontAngle','italic');
title('X-Y Plane View','FontName','Times','FontSize',24,'FontWeight','bold');
 
figure(4)
mesh(x1,x2,fn_tot);view(90,0);colorbar;set(gca,'FontSize',12);
xlabel('x_2','FontName','Times','FontSize',20,'FontAngle','italic');
ylabel('x_1','FontName','Times','FontSize',20,'FontAngle','italic');
zlabel('f(X)','FontName','Times','FontSize',20,'FontAngle','italic');
title('X-Z Plane View','FontName','Times','FontSize',24,'FontWeight','bold');
 
figure(5)
mesh(x1,x2,fn_tot);view(0,0);colorbar;set(gca,'FontSize',12);
xlabel('x_2','FontName','Times','FontSize',20,'FontAngle','italic');
ylabel('x_1','FontName','Times','FontSize',20,'FontAngle','italic');
zlabel('f(X)','FontName','Times','FontSize',20,'FontAngle','italic');
title('Y-Z Plane View','FontName','Times','FontSize',24,'FontWeight','bold');

Click here to download m-file

VI. References:

[1] Xin-She Yang, "Firefly Algorithm, Stochastic Test Functions and Design Optimisation", International Jurnal of Bio-Inspired Computation, vol. 2, no. 2, pp. 78–84, March 2010.
[2] Xin-She Yang, Engineering Optimization: An Introduction with Metaheuristic Applications. Hoboken, New Jersey: John Wiley & Sons Inc, 2010.
[3] M. Jamil and X. S. Yang, "A Literature Survey of Benchmark Functions for Global Optimization Problems," International Journal of Mathematical Modelling and Numerical Optimisation, vol. 4, no. 2, pp. 150–194, Aug. 2013.
[4] A. Gavana, "Test Functions Index," Feb. 2013, [Accessed April 01, 2013]. [Online]. Available: http://infinity77.net/global_optimization/test_functions.html
[5] Ali R. Alroomi, "The Farm of Unconstrained Benchmark Functions," University of Bahrain, Electrical and Electronics Department, Bahrain, Oct. 2013. [Online]. Available: http://www.al-roomi.org/cv/publications