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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');

Click here to download m-file

\(\bullet\) 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');

Click here to download m-file

 

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
Tagged under: unconstrained, benchmark functions, test functions, optimization, Evolutionary Algorithms, MATLAB, Meta-heuristic, Population-Based, Modern Optimization, Multivariate, Quartic Function, 3D model, u3d, Nature-Inspired, n-dimensional, Quartic, 4th De Jong, 4th De Jong Function,