Friedman's Function

I. Mathematical Expression:

$$f(X)=10 \sin\left(\pi x_1 x_2\right)+20\left(x_3-0.5\right)^2+10x_4+5x_5$$

where:

\(\bullet\) \(-1.5\leq x_i \leq 1.5\) , \(i=1,2,\cdots,5\)

\(\bullet\) Infinite number of global minimum: \(f_{min}(x^*)=-27.5\) (based on the given search space)

\(\bullet\) The optima can be analytically determined as:

\( \ \ \ \ \ \ \rhd\) For \(x_1 \ \rightarrow \ x_2\): any values that satisfy \(\sin(y^*)=-1 \ \rightarrow \ y=\pi x_1^* x_2^*\)

\( \ \ \ \ \ \ \rhd\) For \(x_3 \ \rightarrow \ x_5\): \(x^*_j=(0.5,-1.5,-1.5)\) , \(j=3,4,5\)

\(\bullet\) The search space is taken from [1]. The user can use any higher domain for increasing the complexity. However, the optima is always satisfied by the above description, where \(x_3=0.5\), \(x_4=x_4^{min}\) and \(x_5=x_5^{min}\) [2].

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:

\(\bullet\) These plots are created based on eliminating \(x_3\), \(x_4\) and \(x_5\) for seeing the effect of \(x_1\) and \(x_5\), because the effect of the other independent variables is fixed with their optimum values that are given above.

IV. Controllable 3D Model:

\(\bullet\) Again, this model is created based on only \(x_1\) and \(x_2\).

- 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:

% Friedman's Function (when removing x3, x4 and x5)
% Range of initial points: -10 <= xj <= 25 , j=1,2 
% Infinite number of global minimum happen when sin(y)=-1 >> where y=pi*x1*x2
% f(x1,x2)=-10 >> f(X)=-27.5 when n=5
% Coded by: Ali R. Alroomi | Last Update: 11 August 2015 | www.al-roomi.org
 
clear
clc
warning off
 
x1min=-1.5;
x1max=1.5;
x2min=-1.5;
x2max=1.5;
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)=10*sin(pi*x1(j)*x2(i));
    end
 
    f_tot(j,:)=f;
 
end
 
figure(1)
meshc(x1,x2,f_tot);colorbar;set(gca,'FontSize',12);shading interp;
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] Robert B. Gramacy, and Herbert K.H. Lee, "Cases for the Nugget in Modeling Computer Experiments," Statistics and Computing, vol. 22, no. 3, pp. 713-722, May 2012.
[2] 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