Monday, 22 June 2015 07:04
Parent Category: Unconstrained
Category: n-Dimensions
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Generalized Rosenbrock's Valley (Banana or 2nd De Jong's) Function

I. Mathematical Expression:

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

where:

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

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

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

$\bullet$ This benchmark function is very popular and has many names as can be seen from its subject. Also, some references add Saddle instead of Valley $\rightarrow$ "Generalized Rosenbrock's Saddle Function" as in [5, 11, 12]. Moreover, this function comes with different variable bounds, like $|x_i|\leq 10$ as in [5] and $|x_i|\leq 2.048$ as in [12]. Furthermore, there are many other functions are based on or extended from this function; see the listed functions. Please, note that the expression of this benchmark function (when $n=2$) is not similar to that of Leon's Function as many studies, by mistake, consider it.

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 Rosenbrock's Valley (Banana or 2nd De Jong'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: 03 June 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)).^2+(1-x1(j)).^2;
    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');