- Parent Category: Unconstrained
- Category: n-Dimensions
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Quartic (or Modified 4th De Jong's) Function
I. Mathematical Expression:
\(\bullet\) Without Noise:
$$f(X)=\sum^n_{i=1} i x_i^4$$
\(\bullet\) With Noise:
$$f(X)=\sum^n_{i=1} i x_i^4+\text{rand}\left[0,1\right)$$
where:
\(\bullet\) \(-1.28\leq x_i\leq 1.28\) , \(i=1,2,\cdots,n\)
\(\bullet\) \(f_{min}(X^*)=0\)
\(\bullet\) \(x^*_i =0\)
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:
\(\bullet\) Without Noise:
\(\bullet\) With Noise:
IV. Controllable 3D Model:
\(\bullet\) Without Noise:
\(\bullet\) With Noise:
- 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:
\(\bullet\) Without Noise:
% Quartic (or Modified 4th De Jong's) Function (without noise)
% Range of initial points: -1.28 < xj < 1.28 , j=1,2,...,n
% Global minima: (x1,x2,...,xn)=0
% f(X)=0
% Coded by: Ali R. Alroomi | Last Update: 07 June 2015 | www.al-roomi.org
clear
clc
warning off
x1min=-1.28;
x1max=1.28;
x2min=-1.28;
x2max=1.28;
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)=x1(j)^4;
% For 2-dimensional plotting
for i=1:length(x2)
fn(i)=f1(j)+2*x2(i)^4;
end
fn_tot(j,:)=fn;
end
figure(1)
plot(x1,f1);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');
\(\bullet\) With Noise:
% Quartic (or Modified 4th De Jong's) Function (with noise)
% Range of initial points: -1.28 < xj < 1.28 , j=1,2,...,n
% Global minima: (x1,x2,...,xn)=0
% f(X)=0
% Coded by: Ali R. Alroomi | Last Update: 07 June 2015 | www.al-roomi.org
clear
clc
warning off
x1min=-1.28;
x1max=1.28;
x2min=-1.28;
x2max=1.28;
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
RAND1=rand;
while RAND1==1
RAND1=rand; % to avoid RAND1=1
end
f1(j)=x1(j)^4+RAND1;
% For 2-dimensional plotting
for i=1:length(x2)
RAND2=rand;
while RAND2==1
RAND2=rand; % to avoid RAND2=1
end
fn(i)=f1(j)+2*x2(i)^4+RAND2;
end
fn_tot(j,:)=fn;
end
figure(1)
plot(x1,f1);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] R. Storn and K. Price, "Differential Evolution - A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces," Journal of Global Optimization, vol. 11, no. 4, pp. 341-359, Dec. 1997.
[2] 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.
[3] E. P. Adorio, "MVF - Multivariate Test Functions Library in C for Unconstrained Global Optimization," Quezon City, Metro Manila, Philippines, Jan. 2005. [Online]. Available: http://geocities.ws/eadorio/mvf.pdf
[4] S. Rahnamayan, H. R. Tizhoosh, and M. M. A. Salama, "A Novel Population Initialization Method for Accelerating Evolutionary Algorithms," Computers & Mathematics with Applications, vol. 53, no. 10, pp. 1605-1614, May 2007.
[5] S. Rahnamayan, H. Tizhoosh, and M. Salama, "Opposition-Based Differential Evolution for Optimization of Noisy Problems," IEEE Congress on Evolutionary Computation, 2006 (CEC 2006), Jul. 2006, pp. 1865-1872.
[6] D. Simon, Evolutionary Optimization Algorithms: Biologically-Inspired and Population-Based Approaches to Computer Intelligence. Hoboken, New Jersey: John Wiley & Sons Inc, 2013.
[7] M. Jamil and X. S. Yang, "A Literature Survey of Benchmark Functions for Global Optimization Problems," International Journal of Mathematical Modelling and Numerical Optimisation, vol. 4, no. 2, pp. 150–194, Aug. 2013.
[8] A. Gavana, "Test Functions Index," Feb. 2013, [Accessed April 01, 2013]. [Online]. Available: http://infinity77.net/global_optimization/test_functions.html
[9] 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
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