Thursday, 30 July 2015 08:04
Parent Category: Unconstrained
Category: n-Dimensions
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Generalized Dixon-Price-Rosenbrock's Function

I. Mathematical Expression:

$$f(X)=\sum_{i=1}^{n-1}\left[100\left(x_{i+1}-x^2_i\right)^8+\left(x_i-1\right)^8\right]$$

where:

$\bullet$ $-30\leq x_i\leq 30$ , $i=1,2,\cdots,n$

$\bullet$ $f_{min}(X^*)=0$

$\bullet$ $x^*_i =1$

$\bullet$ The original form of this benchmark function is given in [1], and it is similar to Rosenbrock's Function except that the brackets have a power of $8$ instead of $2$. This is why we call it Dixon-Price-Rosenbrock's Function. Based on that, we can consider the other suggested side constraints of Rosenbrock's Function to be used for this new benchmark function. Although it is expressed as a $2-$dimensional problem in [1], the generalized version is expressed in [2] for any higher dimension.

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:

% Generalized Dixon-Price-Rosenbrock's Function
% Range of initial points: -2.048 < xj < 2.048 , j=1,2,...,n
% Some papers take the range as: -5 <= xj <= 10 or -30 <= xj <= 30
% Global minima: (x1,x2,...,xn)=1
% f(X)=0
% Coded by: Ali R. Alroomi | Last Update: 29 May 2015 | www.al-roomi.org
 
clear
clc
warning off
    
x1min=-2.048;
x1max=2.048;
x2min=-2.048;
x2max=2.048;
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)=100*(x2(i)-x1(j)*x1(j)).^8+(1-x1(j)).^8;
    end
    
    f_tot(j,:)=f;
 
end 
 
% 1-dimensional plot is not applicable with this benchmark function
 
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');