Xin-She Yang's Function No.02

I. Mathematical Expression:

$$f(X)=\left(\sum^n_{i=1} \left|x_i\right|\right)\cdot e^{\displaystyle -\sum^n_{i=1}x^2_i}$$

where:

\(\bullet\) \(-10\leq x_i \leq 10\) , \(i=1,2,\cdots,n\)

\(\bullet\) For \(n=1\):

\( \ \ \ \ \ \ \rhd\) It has two maximum: \(f_{max}(x^*)=0.428881942480354\)

\( \ \ \ \ \ \ \rhd\) \(x^*=\pm 0.707106781903310\) (determined by us using MapleSoft 2015)

\(\bullet\) For \(n=2\):

\( \ \ \ \ \ \ \rhd\) It has four maximum: \(f_{max}(X^*)=\frac{\displaystyle 1}{\displaystyle \sqrt{e^1}}=0.606530659712633\)

\( \ \ \ \ \ \ \rhd\) \(x^*_i=\pm \frac{\displaystyle 1}{\displaystyle 2}\)

\(\bullet\) This function is shown in [1, 2], but it is mistakenly skipped in [3, 4] and other papers in the literature. Actually, it comes before Xin-She Yang's Function No.01 in [1], but because the author said that the last one is his first proposed benchmark function, so this benchmark function is called Xin-She Yang's Function No.02 [5].

\(\bullet\) Although, there is one unique global minima at the origin (i.e., \(f_{min}(X^*)=0 \ @ \ x^*_i=0\)), there are infinite number of almost global minimum with so small errors (maybe less than \(10^{-86}\)). Thus, this function is not recommended to be used for minimization unless inverting it by using \(f_{new}(X)=-f_{old}(X)\) [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 # 2
% Range of initial points: -10 <= xj <= 10 , j=1,2,...,n
% For n=1:
% Two global maximum: x=+-0.707106781903310
% f(x)=0.428881942480354 (determined by Maple 2015)
% For n=2:
% Four global maximum: (x1,x2)=(+-0.5,+-0.5)
% f(X)=1/sqrt(exp(1))=0.606530659712633
% Coded by: Ali R. Alroomi | Last Update: 12 August 2015 | www.al-roomi.org
 
% Although, there is one unique global minima at the origin, there are 
% infinite number of almost global minimum with so small errors 
% Thus, this function is not recommended to be used for minimization 
% unless inverting it by using f_new(X)=-f_old(X)

    
clear
clc
warning off
 
x1min=-10;
x1max=10;
x2min=-10;
x2max=10;
R=500; % 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)=abs(x1(j))*exp(-x1(j)^2);
    
    % For 2-dimensional plotting
    for i=1:length(x2)
        fn(i)=(abs(x1(j))+abs(x2(i)))*exp(-x1(j)^2-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