Wednesday, 24 June 2015 10:42
Parent Category: Unconstrained
Category: n-Dimensions
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Needle-Eye Function

I. Mathematical Expression:

$$f(X)=\begin{cases}
1 & \text{ if } \left|x_i\right|\geq eye \ \forall \ i \\
\sum^n_{i=1} \left(100+\left|x_i\right|\right) t_i & \text{ otherwise }
\end{cases}$$

where:

$\bullet$ $eye$ is a small value (it is taken as $0.0001$ in [1, 2])

$\bullet$ $t_i=\begin{cases}
1 & \text{ if } \left|x_i\right| > eye \\
0 & \text{ otherwise }
\end{cases}$

$\bullet$ $-10\leq x_i\leq 10$ , $i=1,2,...,n$ [2]

$\bullet$ Infinite number of global minimum: $f_{min}(X^*)=1$

$\bullet$ Based on our observations, we have found the following:

$\ \ \ \ \ \Rightarrow$ For $n=1$: Infinite number of global minimum happens when $x \in [-1,1]$

$\ \ \ \ \ \Rightarrow$ For $n=2$: Infinite number of global minimum happens when $x_1$ and $x_2$ approach $0$

$\bullet$ This function becomes hard to be minimized as $eye \downarrow$ and $n \uparrow$ [1].

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:

% Needle-Eye Function
% Range of initial points: -10 <= xj <= 10 , j=1,2,...,n
% For n=1:
% Infinite global minimum: any value between -1 <= xj <= 1
% f(x)=1
% For n=2:
% Infinite global minimum: as per x1 and x2 approach 0
% f(x1,x2)=1
% Coded by: Ali R. Alroomi | Last Update: 03 June 2015 | www.al-roomi.org
 
% Minimization of this function becomes more difficult with smaller eye and larger n (dimension)
 
clear
clc
warning off
 
eye=0.0001;
 
x1min=-10;
x1max=10;
x2min=-10;
x2max=10;
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
    if abs(x1(j)) <= eye
        f1(j)=1;
    else
        if abs(x1(j)) > eye
            t1=1;
        else
            t1=0;
        end
        f1(j)=(100+abs(x1(j)))*t1;
    end
    
    % For 2-dimensional plotting
    for i=1:length(x2)
        
        if abs(x1(j)) <= eye && abs(x2(i)) <= eye
            fn(i)=1;
        else            
            if abs(x2(i)) > eye
                t2=1;
            else
                t2=0;
            end
            fn(i)=(100+abs(x1(j)))*t1+(100+abs(x2(i)))*t2;
        end
        
    end
 
    fn_tot(j,:)=fn;
 
end
 
figure(1)
plot(x1,f1);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);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(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] S. Mishra, "Repulsive Particle Swarm Method on Some Difficult Test Problems of Global Optimization," Shillong, India, Oct. 2006. [Online]. Available: http://mpra.ub.uni-muenchen.de/1742/1/MPRA_paper_1742.pdf
[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