Wayburn-Seader's Function No.03

I. Mathematical Expression:

$$f(X)=\frac{2}{3}x_1^3-8x_1^2+33x_1-x_1x_2+5+\left[\left(x_1-4\right)^2+\left(x_2-5\right)^2-4\right]^2$$

where:

\(\bullet\) \(-500\leq x_i\leq 500\) , \(i=1,2\)

\(\bullet\) \(f_{min}(X^*)=19.105879794567979\)

\(\bullet\) \(x^*_i =(5.146896745324582,6.839589743000071)\)

\(\bullet\) These optimal values have been obtained by us using MapleSoft 2015. For ensuring convergence to the global optimal location, the good initial point has been determined by the biogeography-based optimization (BBO) algorithm, which is coded in MATLAB 2011a.

\(\bullet\) Based on [2], the original reference is [1]. However, it has been found that this benchmark function is created by combining a function with its equality constraint (see page 21 in [1]):

$$P(X)=\frac{2}{3}x_1^3-8x_1^2+33x_1-x_1x_2+5$$

\(\hspace{2.5cm}\text{Subjected to}\)

$$h(X)=\left(x_1-4\right)^2+\left(x_2-5\right)^2-4=0$$

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:

% Wayburn-Seader's Function # 3
% Range of initial points: -500 <= xj <= 500 , j=1,2,...,n
% Global minima: (x1,x2)=(5.146896745324582,6.839589743000071)
% f(X)=19.105879794567979
% 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)=2*x1(j).^3/3-8*x1(j).^2+33*x1(j)-x1(j)*x2(i)+5+...
            ((x1(j)-4).^2+(x2(i)-5).^2-4).^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