Tuesday, 16 June 2015 08:39
Parent Category: Unconstrained
Category: 2-Dimensions
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H1 Function

I. Mathematical Expression:

$$f(X)=\frac{\displaystyle \sin\left(x_1-\frac{x_2}{8}\right)^2+\sin\left(x_2+\frac{x_1}{8}\right)^2}{\displaystyle \sqrt{\left(x_1-\frac{36}{13}\pi\right)^2+\left(x_2-\frac{28}{13}\pi\right)^2}+1}$$

where:

$\bullet$ $-25\leq x_i \leq 25$ , $i=1,2$

$\bullet$ $f_{max}(X^*) = 2.0$

$\bullet$ $x^*_i = (\frac{36}{13}\pi,\frac{28}{13}\pi)$

$\bullet$ The original reference of this function is [1]. The fractions of these two decision variables at the optimal point are rounded with four decimal places in [2], and hence the optimal cannot be reached. Based on that, I have adjusted these two elements with the exact values, so that the optimal point of $f_{max} = 2.0$ can be satisfied.

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:

% H1 Function
% Range of initial points: -25 <= xj <= 25 , j=1,2 
% Global maxima: (x1,x2)=(36*pi/13,28*pi/13)
% f(x1,x2)=2.0
% Coded by: Ali R. Alroomi | Last Update: 16 June 2015 | www.al-roomi.org

clear
clc
warning off

x1min=-25;
x1max=25;
x2min=-25;
x2max=25;
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)
        num=(sin(x1(j)-x2(i)/8)).^2+(sin(x2(i)+x1(j)/8)).^2;
        den=sqrt((x1(j)-(36*pi/13)).^2+(x2(i)-(28*pi/13)).^2)+1;
        f(i)=num/den;
    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');