Wednesday, 17 June 2015 12:22
Parent Category: Unconstrained
Category: n-Dimensions
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Exponential Function

I. Mathematical Expression:

$$f(X)=\text{exp}\left(-0.5\sum^{n}_{i=1} x^2_i\right)$$

where:

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

$\bullet$ $f_{max}(X^*)=1$

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

$\bullet$ It has to be known that the global optima of this benchmark function is its maxima not minima as given in [1]. Yes, the given function can be used with minimization algorithms by adding $-ve$ sign as $f^{new}(X)=-f^{old}(X)$, so that the minima becomes negative maxima (i.e., $f_{min}(X^*)=-1$), but this is an another issue.

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:

% Exponential Function
% Range of initial points: -1 <= xj <= 1 , j=1,2 ,...,n
% Global maxima: (x1,x2,...,xn)=0
% f(X)=1
% 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
    f1(j)=exp(-0.5*x1(j).^2);
    
    % For 2-dimensional plotting
    for i=1:length(x2)
        fn(i)=exp(-0.5*(x1(j).^2+x2(i).^2));
    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');