Tuesday, 16 June 2015 11:42
Parent Category: Unconstrained
Category: 2-Dimensions
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Price's Function No.04

I. Mathematical Expression:

$$f(X)=\left(2x_1^3x_2-x_2^3\right)^2+\left(6x_1-x_2^2+x_2\right)^2$$

where:

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

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

$\bullet$ $x^*_i =\{(0, 0), (2, 4), (1.464352119663698, -2.506012760781662)\}$

$\bullet$ This function is taken from [1]. Although [1] cited [2] as a reference for this function, we don't find that function in [2]. Moreover, [3] uses reduced side constraints, $X\in[-50,50]$, instead of $X\in[-500,500]$. Also, its 3D plot is similar to Beale's 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:

% Price's Function # 4
% Range of initial points: -500 <= xj <= 500 , j=1,2 
% Three global minimum: (x1,x2)=(0,0),(2,4),(1.464352119663698,-2.506012760781662)
% f(x1,x2)=0
% Coded by: Ali R. Alroomi | Last Update: 15 June 2015 | www.al-roomi.org

clear
clc
warning off

x1min=-4.500;
x1max=4.500;
x2min=-4.500;
x2max=4.500;
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*x2(i)-x2(i).^3).^2+(6*x1(j)-x2(i).^2+x2(i)).^2;
    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');