- Parent Category: Unconstrained
- Category: n-Dimensions
- Hits: 3388
M (or Hyper-Grid) Function
I. Mathematical Expression:
$$f(X)=-\frac{\displaystyle 1}{\displaystyle n}\sum^n_{i=1} \sin^{\alpha}\left(c \pi x_i\right)$$
where:
\(\bullet\) \(c=5\) as in [1] or \(3\) as in [2], and \(\alpha=6\)
\(\bullet\) \(0\leq x_i \leq 1\) , \(i=1,2,\cdots,n\)
\(\bullet\) It has \(c^n\) global minimum: \(f_{min}(x^*)=-1\)
\(\bullet\) \(x^*_i=\frac{\displaystyle m \sin^{-1}\left(1\right)}{\displaystyle c \pi}\) , \(m=1,3,5,\cdots\)
\(\bullet\) This benchmark function is taken from [1], 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 (\(f_{min}=-1\))
\(\therefore\) \(-\frac{\displaystyle 1}{\displaystyle n}\Big[\sin^{\alpha}\left(c \pi x_1\right)+\sin^{\alpha}\left(c \pi x_2\right)+\cdots+\sin^{\alpha}\left(c \pi x_n\right)\Big]=-1\)
\(\sin^{\alpha}\left(c \pi x_1\right)+\sin^{\alpha}\left(c \pi x_2\right)+\cdots+\sin^{\alpha}\left(c \pi x_n\right)=n\)
Take \(x_1=x_2=\cdots=x_n=x\), so:
\(n\sin^{\alpha}\left(c \pi x\right)=n\)
\(\sin^{\alpha}\left(c \pi x\right)=1\)
\(\sin\left(c \pi x\right)=\sqrt[\alpha]{1}=1\)
\(c \pi x=\sin^{-1}\left(1\right)\)
\(\Rightarrow \ \therefore\) \(x_i=\frac{\displaystyle m \sin^{-1}\left(1\right)}{\displaystyle c \pi}\) , \(m=1,3,5,\cdots\) (this sequence is valid \(\iff \ \alpha\) is even number)
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:
% M (or Hyper-Grid) Function
% Range of initial points: 0 <= xj <= 1 , j=1,2,...,n
% It has c^n global minimum: xj=m*asin(1)/(c*pi) , m=1,3,5,...
% f(X)=-1
% Coded by: Ali R. Alroomi | Last Update: 17 August 2015 | www.al-roomi.org
clear
clc
warning off
alpha=6;
c=5;
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(3*pi*x1(j))^alpha;
% For 2-dimensional plotting
for i=1:length(x2)
fn(i)=-0.5*(sin(c*pi*x1(j))^alpha+sin(c*pi*x2(i))^alpha);
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');
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