- Parent Category: Unconstrained
- Category: 2-Dimensions
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Storn's Function
I. Mathematical Expression:
$$f(X)=10^mx^2_1+x^2_2-\left(x^2_1+x^2_2\right)^2+10^{-m}\left(x^2_1+x^2_2\right)^4$$
where:
\(\bullet\) \(x_{min}\leq x_i\leq x_{max}\) , \(i=1,2\)
\(\bullet\) It has two global minimum as:
\(\bullet\) For \(m=1:\) \(\left[x_{min},x_{max}\right]=[-2,2]\), \(f_{min}(X^*)=-4.07461605632581\times10^{-1}\), \(x^*_i=\left(0,\pm1.386952327146511\right)\)
\(\bullet\) For \(m=2:\) \(\left[x_{min},x_{max}\right]=[-4,4]\), \(f_{min}(X^*)=-1.8058696657349238\times10^{1}\), \(x^*_i=\left(0,\pm2.608906424592038\right)\)
\(\bullet\) For \(m=3:\) \(\left[x_{min},x_{max}\right]=[-8,8]\), \(f_{min}(X^*)=-2.277657499670953\times10^{2}\), \(x^*_i=\left(0,\pm4.701739810796703\right)\)
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:
\(\bullet\) For \(m=1\):
\(\bullet\) For \(m=2\):
\(\bullet\) For \(m=3\):
IV. Controllable 3D Model:
\(\bullet\) The following model has been created when \(m=1\):
- 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:
% Storn's Function
% Range of initial points: min <= xj <= max , j=1,2
% It has two global minimum:
% @ min=-2 and max=2: (x1,x2)=(0,-1.386952327146511),(0,1.386952327146511) >> m=1 and f(x1,x2)=-0.407461605632581
% @ min=-4 and max=4: (x1,x2)=(0,-2.608906424592038),(0,2.608906424592038) >> m=2 and f(x1,x2)=-18.058696657349238
% @ min=-8 and max=8: (x1,x2)=(0,-4.701739810796703),(0,4.701739810796703) >> m=3 and f(x1,x2)=-227.7657499670953
% Coded by: Ali R. Alroomi | Last Update: 23 March 2015 | www.al-roomi.org
clear
clc
warning off
x1min=-2; % it can also be -4 or -8
x1max=2; % it can also be 4 or 8
x2min=-2; % it can also be -4 or -8
x2max=2; % it can also be 4 or 8
R=1500; % steps resolution
x1=x1min:(x1max-x1min)/R:x1max;
x2=x2min:(x2max-x2min)/R:x2max;
m=1; % it can also be 2 or 3
for j=1:length(x1)
for i=1:length(x2)
f(i)=(10.^m)*x1(j).^2+x2(i).^2-(x1(j).^2+x2(i).^2).^2+(10.^-m)*(x1(j).^2+x2(i).^2).^4;
end
f_tot(j,:)=f;
end
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] M. F. P. Costa, F. P. Fernandes, and E. M. da G. P. Fernandes, "A Deterministic-Stochastic Method for Nonconvex MINLP problems," in 2nd International Conference on Engineering Optimization, Sept. 2010, pp. 1-10.
[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
