A+ A A-

I. Mathematical Expression:

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

where:

\(\bullet\) \(-100\leq x_i \leq 100\) , \(i=1,2\)

\(\bullet\) \(f_{min}(X^*) = 0\)

\(\bullet\) \(x^*_i = 0\)

 

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:

% Schaffer's Function # 6
% Range of initial points: -100 <= xj <= 100 , j=1,2
% Some papers, range of initial points: -10 <= xj <= 10 , j=1,2
% Global minima: (x1,x2)=(0,0)
% f(x1,x2)=0
% Coded by: Ali R. Alroomi | Last Update: 24 March 2015 | www.al-roomi.org

clear
clc
warning off
   
x1min=-100;
x1max=100;
x2min=-100;
x2max=100;
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)=0.5+(((sin(sqrt(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

 

VI. References:

[1] M. M. Ali, C. Khompatraporn, and Z. B. Zabinsky, "A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems," Journal of Global Optimization, vol. 31, no. 4, pp. 635-672, Apr. 2005.
[2] S. Mishra, "Some New Test Functions for Global Optimization and Performance of Repulsive Particle Swarm Method," Shillong, India, Aug. 2006. [Online]. Available: http://mpra.ub.uni-muenchen.de/2718/1/MPRA_paper_2718.pdf
[3] E. P. Adorio, "MVF - Multivariate Test Functions Library in C for Unconstrained Global Optimization," Quezon City, Metro Manila, Philippines, Jan. 2005. [Online]. Available: http://geocities.ws/eadorio/mvf.pdf
[4] R. Chelouah and P. Siarry, "Tabu Search Applied to Global Optimization," European Journal of Operational Research, vol. 123, no. 2, pp. 256-270, Jun. 2000.
[5] Ali R. Alroomi, "The Farm of Unconstrained Benchmark Functions," University of Bahrain, Electrical and Electronics Department, Bahrain, Oct. 2013. [Online]. Available: http://www.al-roomi.org/cv/publications