Wednesday, 17 June 2015 09:45
Parent Category: Unconstrained
Category: n-Dimensions
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Cosine Mixture Function

I. Mathematical Expression:

$$f(X)=0.1\sum_{i=1}^{n}\cos\left(5 \pi x_i\right)-\sum_{i=1}^{n}x^2_i$$

where:

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

$\bullet$ $f_{max}(X^*)=0.1$ for $n=1$ , $f_{max}(X^*)=0.2$ for $n=2$ , $f_{max}(X^*)=0.3$ for $n=3$, $f_{max}(X^*)=0.4$ for $n=4$, etc...

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

$\bullet$ It seems that the global maxima is always equal to $\frac{n}{10}$. But this is not true, because when $n$ is large the global maxima will drift away from $\frac{n}{10}$.

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:

% Cosine Mixture Function
% Range of initial points: -1 <= xj <= 1 , j=1,2,...,n
% Global maxima: (x1,x2,...,xn)=0
% f(X)=0.1 for n=1 , f(X)=0.2 for n=2 , f(X)=0.3 for n=3 & f(X)=0.4 for n=4 
% It seems that the global mixima is equal to (n/10). But this is not true,
% because when n is large the global maxima will drift from (n/10)
% Coded by: Ali R. Alroomi | Last Update: 03 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)=0.1*cos(5*pi*x1(j))-x1(j)^2;
    
    % For 2-dimensional plotting
    for i=1:length(x2)
        fa(i)=0.1*cos(5*pi*x1(j))+0.1*cos(5*pi*x2(i));
        fb(i)=x1(j)^2+x2(i)^2;
        fn(i)=fa(i)-fb(i);
    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');