Mishra's Function No.11 (or AMGM Function)

I. Mathematical Expression:

$$f(X)=\left[\frac{1}{n} \sum^{n}_{i=1} \left|x_i\right| - \left(\prod^n_{i=1} \left|x_i\right|\right)^{\frac{1}{n}}\right]^2$$

where:

\(\bullet\) \(0\leq x_i\leq 10\) , \(i=1,2,\cdots,n\)

\(\bullet\) Infinite number of global minimum: \(f_{min}(X^*)=0\)

\(\bullet\) \(x^*_i \ \rightarrow x_1 = x_2 = \cdots = x_n\) and all are non-negative (if \(X_{min}<0\))

\(\bullet\) Its name is "AMGM Function" as in [1]; where AMGM is an acronym of Arithmetic Mean-Geometric Mean. However, this function is called "Mishra's Function No.11" in [2, 3]. Also, we found that this function appeared twice in [3], one with name "AMGM Function" and the other with "Mishra's Function No.11". Based on that, and to clarify this hidden point, we call this function as "Mishra's Function No.11 (or AMGM Function)".

\(\bullet\) Although, from its expression, it shows that it can accept even \(n=1\), this will cause \(f(x)=0\). Thus, from a practical side, this function should not be used with \(n=1\).

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:

% Mishra's # 11 or AMGM Function
% Range of initial points: 0 <= xj <= 10 , j=1,2,...,n
% Global minima: x1=x2=x3=...=xn and all are non-negative
% f(x1,x2,...,xn)=0
% Coded by: Ali R. Alroomi | Last Update: 02 June 2015 | www.al-roomi.org
  
clear
clc
warning off  
  
n=2;  
 
x1min=-10;
x1max=10;
x2min=-10;
x2max=10;
R=1500; % steps resolution
x1=x1min:(x1max-x1min)/R:x1max;
x2=x2min:(x2max-x2min)/R:x2max;
    
for j=1:length(x1)
    
    x1(j)=abs(x1(j));
    
    for i=1:length(x2)       
        x2(i)=abs(x2(i));
        S=x1(j)+x2(i);
        P=x1(j)*x2(i);
        f(i)=((S/n)-((P)^(1/n)))^2;
    end
    
    f_tot(j,:)=f;
 
end
 
% 1-dimensional plot is not allowed 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');

Click here to download m-file

VI. References:

[1] S. Mishra, "Repulsive Particle Swarm Method on Some Difficult Test Problems of Global Optimization," Shillong, India, Oct. 2006. [Online]. Available: http://mpra.ub.uni-muenchen.de/1742/1/MPRA_paper_1742.pdf 
[2] 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.
[3] A. Gavana, "Test Functions Index," Feb. 2013, [Accessed April 01, 2013]. [Online]. Available: http://infinity77.net/global_optimization/test_functions.html
[4] 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