Monday, 17 August 2015 11:10
Parent Category: Unconstrained
Category: n-Dimensions
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L (or F2) Function

I. Mathematical Expression:

$$f(X)=-\prod^n_{i=1}\sin^k\left(l_1 \pi x_i + l_2\right) \cdot e^{\displaystyle -l_3\left(\frac{x_i-l_4}{l_5}\right)^2}$$

where:

$\bullet$ $k=6$, $l_1=5.1$, $l_2=0.5$, $l_3=4\ln(2)$, $l_4=0.066832364099628$, and $l_5=0.64$

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

$\bullet$ $f_{min}(x^*)=-1$

$\bullet$ $x^*_i=l_4$

$\bullet$ It can be noticed that the generated 3D-plot has some similarity with Keane's Function.

$\bullet$ This benchmark function is taken from [1, 2], but without addressing an enough details about the global solution. Thus, it is re-analyzed and the analytical solution is given as follows [3]:

$\because$ The global minima is known ($f_{min}=-1$)

$\because$ There is a product operator and $X\geq 1$

$\therefore$ The minimum value that this function can reach happens when both terms are equal to $1$:

$\blacksquare$ $\sin^k\left(l_1 \pi x_i + l_2\right)=1$ $\rightarrow$ Because $k$ is even, so this term is equal to $1$ if its angle $l_1 \pi x_i + l_2=\pm 1$

$\blacksquare$ $e^{\displaystyle -l_3\left(\frac{x_i-l_4}{l_5}\right)^2} \ \iff \ x_i=l_4$

However, the given value of $l_4$ in [1] is $0.0667$. Therefore, when $x_i=l_4$ $\rightarrow$ the first term will not be equal to $1$ anymore

Thus, either the optima is bigger than $-1$ or the exact value of $l_4$ is rounded

The second approach is taken in this analysis, so the corrected $l_4$ is:

$\tilde{l}_4=\frac{\displaystyle \sin^{-1}\left(1\right)-l_2}{\displaystyle l_1 \pi}$

$\Rightarrow \ \therefore$ $x_i=\tilde{l}_4=0.066832364099628$

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:

% L (or F2) Function
% Range of initial points: 0 <= xj <= 1 , j=1,2,...,n
% Global minima: (x1,x2,...,xn)= L4;
% f(X)=-1
% Coded by: Ali R. Alroomi | Last Update: 17 August 2015 | www.al-roomi.org
    
clear
clc
warning off
 
% coefficients
L1=5.1;
L2=0.5;
L3=4*log(2);
L4=0.066832364099628;%0.0667;
L5=0.64;
k=6;
 
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
    f1(j)=-sin(L1*pi*x1(j)+L2)^k*exp(-L3*((x1(j)-L4)/L5)^2);
    
    % For 2-dimensional plotting
    for i=1:length(x2)
        fn(i)=-(sin(L1*pi*x1(j)+L2)^k*exp(-L3*((x1(j)-L4)/L5)^2)*...
            sin(L1*pi*x2(i)+L2)^k*exp(-L3*((x2(i)-L4)/L5)^2));
    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);shading interp;
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] Ofer M. Shir, "Niching in Derandomized Evolution Strategies and its Applications in Quantum Control; A Journey from Organic Diversity to Conceptual Quantum Designs," Ph.D. Thesis, ISBN: 978-90-6464-256-2, Leiden University, Leiden, The Netherlands, 2008, [Accessed Aug. 16, 2015]. [Online]. Available: http://cs.telhai.ac.il/~ofersh/OferMichaelShir_PhDmanuscript.pdf
[2] Ofer M. Shir, and Thomas Bäck, "Niche Radius Adaptation in the CMA-ES Niching Algorithm," in Parallel Problem Solving from Nature - PPSN IX (ser. Lecture Notes in Computer Science, vol. 4193), Thomas Philip Runarsson et al. Berlin, Heidelberg: Springer Berlin Heidelberg, 2006, pp. 142-151.
[3] 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