- Parent Category: Unconstrained
- Category: n-Dimensions
- Hits: 3258
Schumer-Steiglitz's Function No.03
I. Mathematical Expression:
$$f(X)=\sum^n_{i=1} a_i x^2_i$$
where:
\(\bullet\) \(a\) is an \(n-\)dimensional vector of randomly chosen coefficients from a uniform distribution on \([0.1,1]\)
\(\bullet\) \(-100\leq x_i\leq 100\) , \(i=1,2,\cdots,n\)
\(\bullet\) \(f_{min}(X^*)=0\)
\(\bullet\) \(x^*_i =0\)
\(\bullet\) There is no side constraints associated with this function in [1]. Because Schumer-Steiglitz's Function No.01 is similar to Sphere Model, so \(x_i \in [-100,100]\) is used for this function too [2].
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:
% Schumer-Steiglitz's Function # 3
% Range of initial points: -100 <= xj <= 100 , j=1,2,...,n (my suggestion)
% Global minimaa: (x1,x2,...,xn)=0
% f(X)=0
% Coded by: Ali R. Alroomi | Last Update: 05 August 2015 | www.al-roomi.org
clear
clc
warning off
x1min=-1;
x1max=1;
x2min=-1;
x2max=1;
R=1000; % 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)=(0.1+0.9*rand)*x1(j)^2;
% For 2-dimensional plotting
for i=1:length(x2)
fn(i)=f1(j)+(0.1+0.9*rand)*x2(i)^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);
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] Michael A. Schumer, and Kenneth Steiglitz, "A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems," IEEE Transactions on Automatic Control, vol. AG-13, no. 3, pp. 270-276, Jun. 1968.
[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
