Wednesday, 06 May 2015 08:39
Parent Category: Unconstrained
Category: 2-Dimensions
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Modified Schaffer's Function No.03

I. Mathematical Expression:

$$f(X)=0.5+\frac{\sin^2\left[\cos\left(\left|x^2_1-x^2_2\right|\right)\right]-0.5}{\left[1+0.001\left(x^2_1+x^2_2\right)\right]^2}$$

where:

$\bullet$ $-100\leq x_i \leq 100$ , $i=1,2$

$\bullet$ Four global minimum are known: $f_{min}(X^*) = 0.001566854526004$

$\bullet$ $x^*_i \approx (0, \pm 1.253114962205510),(\pm 1.253114962205510, 0)$

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:

% Modified Schaffer's Function # 3
% Range of initial points: -100 <= xj <= 100 , j=1,2
% Some papers, range of initial points: -10 <= xj <= 10 , j=1,2
% 4 global mainimum are known: (x1,x2)=(0,+-1.253114962205510),(+-1.253114962205510,0)  
% f(x1,x2)=0.001566854526004
% Coded by: Ali R. Alroomi | Last Update: 25 March 2015 | www.al-roomi.org

% The correct equation is taken from: 
% http://mpra.ub.uni-muenchen.de/2718/1/MPRA_paper_2718.pdf 

% Also, the given information is not precise and with only one global optima
% I re-analyzed this benchmark again and the information is corrected
    
clear
clc
warning off
    
x1min=-10;
x1max=10;
x2min=-10;
x2max=10;
R=600; % 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)=0.5+(((sin(cos(abs(x1(j).^2-x2(i).^2)))).^2-0.5)/(1+0.001*(x1(j).^2+x2(i).^2)).^2);                           
    end
        
    f_tot(j,:)=f;

end

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