Wednesday, 17 June 2015 11:29
Parent Category: Unconstrained
Category: n-Dimensions
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EX3 (Csendes' or Infinity) Function

I. Mathematical Expression:

$$f(X)=\begin{cases}
\sum_{i=1}^{n}x^6_i\left[\sin\left(\frac{1}{x_i}\right)+2\right] & \text{ if } \prod^{n}_{i=1} x_i \neq 0 \\
0 & \text{ otherwise }
\end{cases}$$

where:

$\bullet$ $-1\leq x_i\leq 1$ as in [1] or $-2\leq x_i\leq 2$ as in [2], $i=1,2,\cdots,n$

$\bullet$ $f_{min}(X^*)=0$

$\bullet$ $x^*_i =0$

$\bullet$ This function was firstly designed by T. Csendes in [1], and thus it is sometimes known as Csendes' Function as in [3, 4]. In [2], this function was represented again with a new name "EX3 Function". Also, this function is called Infinity Function as in [4]. Please note that there are some typo-errors in the given expression in [3, 4], where at $x_i=0$ the dependent variable will be equal to $0$; otherwise it will go to infinity.

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:

% Csendes' (EX3 or Infinity) Function
% Range of initial points: -1 <= xj <= 1 , j=1,2,...,n
% Other proposed side constraints: -2 <= xj <= 2 
% Infinite number of global minimum happen if at least one element xj=0
% f(X)=0
% Coded by: Ali R. Alroomi | Last Update: 17 June 2015 | www.al-roomi.org

clear
clc
warning off

x1min=-1;
x1max=1;
x2min=-1;
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
    if x1(j)~=0
        f1(j)=(sin(1/x1(j))+2)*x1(j)^6;
    else
        f1(j)=0;
    end
    
    % For 2-dimensional plotting
    for i=1:length(x2)
        if x1(j)*x2(i)~=0
            fn(i)=f1(j)+(sin(1/x2(i))+2)*x2(i)^6;
        else
            fn(i)=0;
        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);
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');