- Parent Category: Unconstrained
- Category: n-Dimensions
- Hits: 13757
Generalized Penalized Function No.01
I. Mathematical Expression:
$$f(X)=\frac{\pi}{n}\times\Bigg\{10\sin^2\left(\pi y_1\right)+\sum_{i=1}^{n-1}\left(y_i-1\right)^2\left[1+10\sin^2\left(\pi y_{i+1}\right)\right]+\left(y_n-1\right)^2\Bigg\}+\sum_{i=1}^{n}u\left(x_i,a,k,m\right)$$
where:
\(\bullet\) \(y_i=1+\frac{1}{4}\left(x_i+1\right) \ \ , \ \ \ u\left(x_i,a,k,m\right)=\begin{cases}
\displaystyle k\left(x_i-a\right)^m & \text{ if } x_i>a \\
0 & \text{ if } -a\leq x_i\leq a \\
\displaystyle k\left(-x_i-a\right)^m & \text{ if } x_i<-a
\end{cases}\)
\(\bullet\) \(a=10, \ \ k=100, \ \ m=4\)
\(\bullet\) \(-50\leq x_i\leq 50\) , \(i=1,2,\cdots,n\)
\(\bullet\) \(f_{min}(X^*)=0\)
\(\bullet\) \(x^*_i =-1\)
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:
% Generalized Penalized Function # 1
% Range of initial points: -50 <= xj <= 50 , j=1,2,...,n
% Global minima: (x1,x2,...,xn)=-1
% f(X)=0
% Coded by: Ali R. Alroomi | Last Update: 10 June 2015 | www.al-roomi.org
clear
clc
warning off
x1min=-5;
x1max=5;
x2min=-5;
x2max=5;
R=1500; % steps resolution
x1=x1min:(x1max-x1min)/R:x1max;
x2=x2min:(x2max-x2min)/R:x2max;
a=10;
k=100;
m=4;
n=2;
for j=1:length(x1)
y1=1+0.25*(x1(j)+1);
if x1(j)>a
u1=k*(x1(j)-a).^m;
elseif x1(j)<(-a)
u1=k*(-x1(j)-a).^m;
else
u1=0;
end
for i=1:length(x2)
y2=1+0.25*(x2(i)+1);
if x2(i)>a
u2=k*(x2(i)-a).^m;
elseif x2(i)<(-a)
u2=k*(-x2(i)-a).^m;
else
u2=0;
end
g=((y1-1).^2)*(1+10*(sin(pi*y2)).^2);
f(i)=(pi/n)*((10*(sin(pi*y1)).^2)+g+(y2-1).^2)+u1+u2;
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');
VI. References:
[1] X. Yao, Y. Liu, and G. Lin, "Evolutionary Programming Made Faster," IEEE Transactions on Evolutionary Computation, vol. 3, no. 2, pp. 82-102, Jul. 1999.
[2] A. K. Qin, V. L. Huang, and P. Suganthan, "Differential Evolution Algorithm with Strategy Adaptation for Global Numerical Optimization," IEEE Transactions on Evolutionary Computation, vol. 13, no. 2, pp. 398-417, April 2009.
[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
