Thursday, 14 May 2015 11:41
Parent Category: Unconstrained
Category: 2-Dimensions
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Rump Function

I. Mathematical Expression:

$$f(X)=\left|\left(333.75-x^2_1\right)x^6_2+x^2_1\left(11 x^2_1 x^2_2 -121x^4_2-2\right)+5.5x^8_2+\frac{x_1}{2 + x_2}\right|$$

where:

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

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

$\bullet$ $x^*_i=(0, 0)$

$\bullet$ Please note that this function has been modified by us in order to satisfy the given optimal point. The function taken from [1] comes without absolute brackets (this means $f(X)$ will go to huge negative values), and the unmodified denominator of the last term is $2 x_2$ (this means we will have "infinite" if $x_2=0$ and $x_1 \neq0$, or "indeterminate" if $x_1=x_2=0$).

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:

% Rump Function
% Range of initial points: -500 <= xj <= 500 , j=1,2
% Global minima: (x1,x2)=(0,0)
% f(x1,x2)=0
% Coded by: Ali R. Alroomi | Last Update: 14 May 2015 | www.al-roomi.org

clear
clc
warning off

x1min=-1;
x1max=1;
x2min=-1;
x2max=1;
R=500; % 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)=abs((333.75-x1(j).^2)*x2(i).^6+x1(j).^2*(11*x1(j).^2*x2(i).^2-121*x2(i).^4-2)+5.5*x2(i).^8+x1(j)/(2+x2(i)));
    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');