Monday, 27 July 2015 10:42
Parent Category: Unconstrained
Category: n-Dimensions
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Deb's Function No.02

I. Mathematical Expression:

$$f(X)=-\frac{1}{n}\sum^n_{i=1}\sin^6\left[5 \pi \left(x^{3/4}_i-0.05\right)\right]$$

where:

$\bullet$ $0\leq x_i\leq 1$ , $i=1,2,\cdots,n$

$\bullet$ Many global minimum: $f_{min}(X^*)=-1$

$\bullet$ These optima are located at points when each $i$th term of $5 \pi \left(x^{3/4}_i-0.05\right)$ is equal to $\frac{\displaystyle m \pi}{\displaystyle 2}$, where $m=1, 3, 5 ,7,9$. Thus, there are $5^n$ optima;

$\bullet$ For $n=1$: $5$ optima when $x=\left(\frac{\displaystyle m}{10}+0.05\right)^{4/3}$

$\bullet$ For $n=2$: $25$ optima when $x_i=\left(\frac{\displaystyle m}{10}+0.05\right)^{4/3}$

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:

% Deb's Function # 2
% Range of initial points: 0 <= xj <= 1 , j=1,2,...,n
% Many global minimum happen when each ith term of (5*pi*(xj^(3/4)-0.05)) 
% is equal to (m*pi/2), where m=1,3,5,7,9
% Thus, there are 10^n optimal points:
% For n=1: 5 optima when x=(m/10+0.05)^(4/3)
% For n=2: 25 optima when xj=(m/10+0.05)^(4/3)
% f(X)=-1
% Coded by: Ali R. Alroomi | Last Update: 27 July 2015 | www.al-roomi.org
    
clear
clc
warning off
 
x1min=0;
x1max=1;
x2min=0;
x2max=1;
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)=-(sin(5*pi*(x1(j)^(3/4)-0.05)))^6;
    
    % For 2-dimensional plotting
    for i=1:length(x2)
        fn(i)=(-1/2)*((sin(5*pi*(x1(j)^(3/4)-0.05)))^6+(sin(5*pi*(x2(i)^(3/4)-0.05)))^6);
    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] 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.
[2] A. Gavana, "Test Functions Index," Feb. 2013, [Accessed April 01, 2013]. [Online]. Available: http://infinity77.net/global_optimization/test_functions.html
[3] 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