Branin's RCOS Function No.02

I. Mathematical Expression:

$$f(X)=\frac{-1}{f_1+f_2+f_3+e}$$

where:

\(\bullet\) \(a=1, \ b=\frac{5.1}{4\pi^2}, \ c=\frac{5}{pi}, \ d=6, \ e=10, \ g=\frac{1}{8\pi}\)

\(\bullet\) \(f_1=a\left(x_2 - b x^2_1 + c x_1 - d\right)^2\)

\(\bullet\) \(f_2=e \left(1-g\right) \ cos\left(x_1\right) \cos\left(x_2\right)\)

\(\bullet\) \(f_3=\log\left(x^2_1 + x^2_2 + 1\right)\)

\(\bullet\) \(-5\leq x_1 \leq 10\) , \(0\leq x_2 \leq 15\)

\(\bullet\) \(f_{min}(X^*) = -0.179891239069905\)

\(\bullet\) \(x^*_i = (-3.196988423389338,12.526257883092258)\)

\(\bullet\) This function is called "Modified Branin's RCOS Function" in [1], while it is called "Branin's RCOS Function No.02" in [2, 3]. Also, we found that the given side constraints in [2, 3] are symmetrical, while in [1] each independent variable has its own lower and upper bounds. Moreover, the function expressions given in [2, 3] are different than that given in [1]. Based on that, we stay on the name "Branin's RCOS Function No.02" for having good indexing, but we use the detailed expression in [1] to construct this function.

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:

% Branin's RCOS Function # 2
% Range of initial points: -5 <= xj <= 10 , 0 <= xj <= 15
% Global minima: (x1,x2)=(-3.196988423389338,12.526257883092258)
% f(x1,x2)=-0.179891239069905
% Coded by: Ali R. Alroomi | Last Update: 26 May 2015 | www.al-roomi.org

clear
clc
warning off
    
x1min=-5;
x1max=10;
x2min=0;
x2max=15;
R=1500; % steps resolution
x1=x1min:(x1max-x1min)/R:x1max;
x2=x2min:(x2max-x2min)/R:x2max;

% constants
a=1;
b=5.1/(4*pi^2);
c=5/pi;
d=6;
e=10;
g=1/(8*pi);
    
for j=1:length(x1)
    
    for i=1:length(x2)
        f1=a*(x2(i)-b*x1(j).^2+c*x1(j)-d)^2;
        f2=e*(1-g)*cos(x1(j))*cos(x2(i));
        f3=log(x1(j).^2+x2(i).^2+1);
        f(i)=-1/(f1+f2+f3+e);
    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] S. Mishra, "Repulsive Particle Swarm Method on Some Difficult Test Problems of Global Optimization," Shillong, India, Oct. 2006. [Online]. Available: http://mpra.ub.uni-muenchen.de/1742/1/MPRA_paper_1742.pdf
[2] M. Jamil and X. S. Yang, "A Literature Survey of Benchmark Functions for Global Optimization Problems," International Journal of Mathematical Modelling and Numerical Optimisation, vol. 4, no. 2, pp. 150–194, Aug. 2013.
[3] A. Gavana, "Test Functions Index," Feb. 2013, [Accessed April 01, 2013]. [Online]. Available: http://infinity77.net/global_optimization/test_functions.html
[4] 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