Wednesday, 24 June 2015 06:05
Parent Category: Unconstrained
Category: n-Dimensions
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Inverted Cosine-Wave Function

I. Mathematical Expression:

$$f(X)=-\sum_{i=1}^{n-1}\Bigg\{e^{\displaystyle\left[\frac{-\left(x^2_i+x^2_{i+1}+0.5x_ix_{i+1}\right)}{8}\right]}\cos \bigg( 4\times\sqrt{x^2_i+x^2_{i+1}+0.5x_ix_{i+1}}\bigg)\Bigg\}$$

where:

$\bullet$ $-5\leq x_i\leq 5$ , $i=1,2,\cdots,n$

$\bullet$ $f_{min}(X^*)=-n+1$

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

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:

% Inverted Cosine Wave Function
% Range of initial points: -5 <= xj <= 5 , j=1,2
% Global maxima: (x1,x2)=0
% f(x1,x2)=-n+1 (for n=2, fmin=-1)
% Coded by: Ali R. Alroomi | Last Update: 23 March 2015 | www.al-roomi.org
 
clear
clc
warning off
 
x1min=-5;
x1max=5;
x2min=-5;
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)
        f(i)=-exp(-(x1(j)^2+x2(i)^2+0.5*x1(j)*x2(i))/8)*cos(4*sqrt(x1(j)^2+x2(i)^2+0.5*x1(j)*x2(i)));
    end
    
    f_tot(j,:)=f;
 
end
 
% 1-dimensional plot is not allowed with this benchmark function
 
figure(1)
meshc(x1,x2,f_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(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