Thursday, 06 August 2015 05:02
Parent Category: Unconstrained
Category: n-Dimensions
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Neumaier's Function No.03 (or Trid's Function)

I. Mathematical Expression:

$$f(X)=\sum_{i=1}^{n}\left(x_i-1\right)^2-\sum_{i=2}^{n}x_i \cdot x_{i-1}$$

where:

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

$\bullet$ $f_{min}(X^*)=-\frac{\displaystyle n(n+4)(n-1)}{\displaystyle 6} \ \rightarrow \ f_{min}(X^*)\Bigg|_{n=15}=-665$

$\bullet$ $x^*_i =i(n+1-i)$

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:

% Neumaier's Function # 3 (or Trid's Function)
% Range of initial points: -n^2 <= xj <= n^2 , j=1,2,...,n
% Global minima: xj=j(n+1-j);
% f(X)=-n(n+4)(n-1)/6 >> for example: if n=15, then fmin=-665
% Coded by: Ali R. Alroomi | Last Update: 06 August 2015 | www.al-roomi.org
    
clear
clc
warning off
    
x1min=-15^2;
x1max=15^2;
x2min=-15^2;
x2max=15^2;
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)=(x1(j)-1)^2+(x2(i)-1)^2-x1(j)*x2(i);
    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');

Click here to download m-file