Adjiman's Function

I. Mathematical Expression:

\(\bullet\) The Adjiman's benchmark function is simply a sine and cosine function as follows:

$$f(X)=\cos(x_1)\sin(x_2)-\frac{x_1}{x^2_2+1}$$

where:

\(\bullet\) \(-5\leq x_i\leq 5\) , \(i=1,2\)  \(\rightarrow\) Some references have different variable bounds

\(\bullet\) \(f_{min}(X^*)=-x^{max}_1\)  \(\rightarrow\) As per above variable bounds, \(f_{min}(X^*)=-5\)

\(\bullet\) \(x^*_i = (x^{max}_1,0)=(5,0)\)

\(\bullet\) The given variable bounds are taken from [1]. Also, there are other variable bounds; for example (\(x_1 \in \left[-1,2\right]\) , \(x_2 \in \left[-1,1\right]\)) as in [2, 3].

\(\bullet\) This function gives one interesting thing, where the global point can always be calculated without worry about the scale of the problem. Here, the global minima is always at \((x_1,x_2)=(x^{max}_1,0)\). For example, \(f_{min}(X)=f(x_1=5,x_2=0)=-5\). Similarly, for the global maxima \(f_{max}\), where it will always be located at \((x_1,x_2)=(-x^{max}_1,0)\). However, if the lower and upper bounds are not similar, as in [2, 3], then this condition may not be valid for \(f_{max}\).

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:

% Adjiman's Function
% Range of initial points: -5 <= xj <= 5 , j=1,2
% Some references: -1 <= x1 <= 2 , -1 <= x2 <= 1 "different global minima"
% Global minima: (x1,x2)=(5,0)
% f(x1,x2)=-5
% Coded by: Ali R. Alroomi | Last Update: 11 May 2015 | www.al-roomi.org

clear
clc
warning off
    
x1min=-5;
x1max=5;
x2min=-5;
x2max=5;
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)
        f(i)=cos(x1(j))*sin(x2(i))-x1(j)/(1+x2(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] Troy Allen Henderson, "A Learning Approach to Sampling Optimization: Applications in Astrodynamics," Ph.D. Dissertation, Texas A&M University, Texas, 2013, [Accessed May. 2, 2015]. [Online]. Available: http://repository.tamu.edu/bitstream/handle/1969.1/151266/HENDERSON-DISSERTATION-2013.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