Wednesday, 29 July 2015 04:23
Parent Category: Unconstrained
Category: n-Dimensions
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Type-I Simple Deceptive Problem

I. Mathematical Expression:

$$f(X)=\Bigg[\frac{\displaystyle 1}{\displaystyle n} \sum^n_{i=1}g_i(x_i)\Bigg]^{\beta}$$

where:

$\bullet$ $\beta$ is a non-linearity factor. It is considered equal to 1 in [1]

$\bullet$ $g_i(x_i)=\begin{cases}
\alpha-x_i & \text{ if } 0 \leq x_i \leq \alpha \\
\frac{\displaystyle x_i-\alpha}{\displaystyle 1-\alpha} & \text{ otherwise }
\end{cases}$

$\bullet$ $\alpha=0.8$

$\bullet$ $0\leq x_i\leq 1$ , $i=1,2,\cdots,n$

$\bullet$ $f_{max}(X^*)=1$ with $2^n-1$ local maximum points

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

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:

% Type-I Simple Deceptive Problem
% Range of initial points: 0 <= xj <= 1 , j=1,2,...,n
% Global maxima: (x1,x2,...,xn)=1 "with (2^n)-1 local maximum"
% f(X)=1
% Coded by: Ali R. Alroomi | Last Update: 27 July 2015 | www.al-roomi.org
    
clear
clc
warning off
 
alpha=0.8;
beta=1;
 
x1min=0;
x1max=1;
x2min=0;
x2max=1;
R=1500; % steps resolution
x1=x1min:(x1max-x1min)/R:x1max;
x2=x2min:(x2max-x2min)/R:x2max;
 
for j=1:length(x1)
    
    % For 1-dimensional plotting
    if x1(j)<=alpha
        g1=alpha-x1(j);
    else
        g1=(x1(j)-alpha)/(1-alpha);
    end
    
    f1(j)=g1^beta;
    
    % For 2-dimensional plotting
    for i=1:length(x2)
        
        if x2(i)<=alpha
            g2=alpha-x2(i);
        else
            g2=(x2(i)-alpha)/(1-alpha);
        end
        
        fn(i)=(0.5*(g1+g2))^beta;
        
    end
 
    fn_tot(j,:)=fn;
 
end
 
figure(1)
plot(x1,f1,'r','LineWidth',2);set(gca,'FontSize',12);
xlabel('x','FontName','Times','FontSize',20,'FontAngle','italic');
ylabel('f(x)','FontName','Times','FontSize',20,'FontAngle','italic');
title('2D View','FontName','Times','FontSize',24,'FontWeight','bold');
 
figure(2)
meshc(x1,x2,fn_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(3)
mesh(x1,x2,fn_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(4)
mesh(x1,x2,fn_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(5)
mesh(x1,x2,fn_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

VI. References:

[1] Hideaki Suzuki, and Hidefumi Sawai, "Chemical Genetic Algorithms --- Coevolution between Codes and Code Translation," in Proceedings of the Eighth International Conference on Artificial Life (Artificial Life VIII), 2002, pp. 164-172. [Online]. Available: http://www.alife.org/alife8/proceedings/sub2029.pdf
[2] Ali R. Alroomi, "The Farm of Unconstrained Benchmark Functions," University of Bahrain, Electrical and Electronics Department, Bahrain, Oct. 2013. [Online]. Available: http://www.al-roomi.org/cv/publications