Tuesday, 16 June 2015 10:22
Parent Category: Unconstrained
Category: 2-Dimensions
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Three-Cylinders Function

I. Mathematical Expression:

$$f(X)=\begin{cases}
1 & \text{ if } r_1\geq 0.75 \\
1.05 & \text{ if } r_2\geq 0.375 \\
1.05 & \text{ if } r_3\geq 0.25 \\
0 & \text{ otherwise }
\end{cases}$$

where:

$\bullet$ $r_1=\sqrt{\left(x_1-3\right)^2+\left(x_2-2\right)^2}$

$\bullet$ $r_2=\sqrt{\left(x_1-4\right)^2+\left(x_2-4\right)^2}$

$\bullet$ $r_3=\sqrt{\left(x_1-1\right)^2+\left(x_2-3\right)^2}$

$\bullet$ $0\leq x_i \leq 5$ , $i=1,2$

$\bullet$ Infinite number of global maximum: $f_{max}(X^*) = 1.05$

$\bullet$ These infinite optimal points are located on the ends of the smaller diameter cylinders whose values are only 5% greater than the local maxima centered on $(3,2)$ and occupying a much larger area.

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:

% Three Cylinders Function
% Range of initial points: 0 <= xj <= 5 , j=1,2 
% Infinite number of global mamimum: located on the ends of the smaller 
% diameter cylinders whose values are only 5% greater than the local maxima
% centered on (3,2) and occupying a much larger area
% f(x1,x2)=1.05
% Coded by: Ali R. Alroomi | Last Update: 04 June 2015 | www.al-roomi.org

clear
clc
warning off

x1min=0;
x1max=5;
x2min=0;
x2max=5;
R=1500; % steps resolution
x1=x1min:(x1max-x1min)/R:x1max;
x2=x2min:(x2max-x2min)/R:x2max;
    
for j=1:length(x1)
    
    for i=1:length(x2)
        r1=sqrt((x1(j)-3).^2+(x2(i)-2).^2);
        r2=sqrt((x1(j)-4).^2+(x2(i)-4).^2);
        r3=sqrt((x1(j)-1).^2+(x2(i)-3).^2);
        if r1 <= 0.75
            f(i)=1;
        elseif r2 <= 0.375
            f(i)=1.05;
        elseif r3 <= 0.25
            f(i)=1.05;
        else
            f(i)=0;
        end
    end

    f_tot(j,:)=f;

end

figure(1)
meshc(x1,x2,f_tot);colorbar;set(gca,'FontSize',12);shading interp;
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(2)
mesh(x1,x2,f_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(3)
mesh(x1,x2,f_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(4)
mesh(x1,x2,f_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] R. A. Formato, "Central Force Optimization: A New Metaheuristic with Applications in Applied Electromagnetics," Progress In Electromagnetics Research, vol. 77, pp. 425-491, 2007. [Online]. Available: http://www.jpier.org/PIER/pier77/33.07082403.Formato.pdf
[2] 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