Mishra's Function No.08 (or Decanomial Function)

I. Mathematical Expression:

$$f(X)=0.001\left(\left|g\left(x_1\right)\right|+\left|h\left(x_2\right)\right|\right)^2$$

where:

\(\bullet\) \(g\left(x_1\right) = x^{10}_1-20x^9_1+180x^8_1-960x^7_1+3360x^6_1-8064x^5_1+13340x^4_1-\)

\(15360x^3_1+11520x^2_1-5120x_1+2624.0\)

\(\bullet\) \(h\left(x_2\right) = x^4_2+12x^3_2+54x^2_2+108x_2+81.0\)

\(\bullet\) \(-10\leq x_i\leq 10\) , \(i=1,2\)

\(\bullet\) \(f_{min}(X^*)=0\)

\(\bullet\) \(x^*_i=(2,-3)\)

\(\bullet\) Its name is "New Decanomial Function" as in [1], while [2] called it "Mishra's Function No.08" since it is a new function by Mishra in [1]. Also, we found that this function appeared twice in [3], one with name "Decanomial Function" and the other with "Mishra's Function No.08". Based on that, and to clarify this hidden point, we call this function as "Mishra's Function No.08 (or Decanomial Function)".

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 # 8 or Decanomial Function
% Range of initial points: -10 <= xj <= 10 , j=1,2 
% Global minima: (x1,x2)=(2,-3)
% f(x1,x2)=0
% Coded by: Ali R. Alroomi | Last Update: 24 March 2015 | www.al-roomi.org

clear
clc
warning off

x1min=0;
x1max=2.5;
x2min=-4;
x2max=-2;
R=1500; % steps resolution
x1=x1min:(x1max-x1min)/R:x1max;
x2=x2min:(x2max-x2min)/R:x2max;

for j=1:length(x1)
    
    for i=1:length(x2)
        % First polynomial
        f1(i)=x1(j).^10-20*x1(j).^9+180*x1(j).^8-960*x1(j).^7+3360*x1(j).^6-8064*x1(j).^5+13340*x1(j).^4-15360*x1(j).^3+11520*x1(j).^2-5120*x1(j)+2624;
        % Second polynomial
        f2(i)=x2(i).^4+12*x2(i).^3+54*x2(i).^2+108*x2(i)+81;
        % overall equation
        f(i)=0.001*(abs(f1(i))+abs(f2(i))).^2;
    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');

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