Tuesday, 18 August 2015 10:42
Parent Category: Unconstrained
Category: 2-Dimensions
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Wayburn-Seader's Function No.02

I. Mathematical Expression:

$$f(X)=\left[1.613-4\left(x_1-0.3125\right)^2-4\left(x_2-1.625\right)^2\right]^2+\left(x_2-1\right)^2$$

where:

$\bullet$ $-500\leq x_i\leq 500$ , $i=1,2$

$\bullet$ It has two global minimum: $f_{min}(X^*)=0$

$\bullet$ $x^*_i =\{(0.200138974728779,1),(0.424861025271221,1)\}$ $\rightarrow$ If $x_2$ is set to $1$, then the analytical solution can be obtained by:

$$x_1=0.3125 \pm \sqrt{\frac{\displaystyle 1.613}{\displaystyle 4}-0.625^2}$$

$\bullet$ Based on [2], the original reference is [1]. However, it has been found that this benchmark function is created by modifying and combining the following two equations (see page 18 in [1]):

$$f_1(x_2)=x_2-1=0$$

$$f_2(X)=1.613-4\left(x_1-0.3125\right)^2-4\left(x_2-1.625\right)^2=0$$

$\bullet$ Also, the above two equations are modified in [1] with absolute brackets as follows:

$$f_1(x_2)=\left|x_2\right|-1=0$$

$$f_2(X)=1.613-4\left(\left|x_1\right|-0.3125\right)^2-4\left(\left|x_2\right|-1.625\right)^2=0$$

$\bullet$ Thus, a new benchmark function (Wayburn-Seader's Function No.04) can be proposed as [3]:

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:

% Wayburn-Seader's Function # 2
% Range of initial points: -500 <= xj <= 500 , j=1,2,...,n
% Two global minima: (x1,x2)={(0.200138974728779,1),(0.424861025271221,1)}
% The analytical solution: set x2=1 and x1=0.3125+-sqrt((1.613/4)-0.625^2)
% f(X)=0
% Coded by: Ali R. Alroomi | Last Update: 18 August 2015 | www.al-roomi.org
    
clear
clc
warning off
 
x1min=-5;
x1max=5;
x2min=-5;
x2max=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)=(1.613-4*(x1(j)-0.3125).^2-4*(x2(i)-1.625).^2).^2+(x2(i)-1).^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] T. L. Wayburn, and J. D. Seader, "Homotopy Continuation Methods for Computer-Aided Process Design," Computers & Chemical Engineering, vol. 11, no. 1, pp. 7-25, 1987.
[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] 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