Wednesday, 18 March 2015 12:10
Parent Category: Unconstrained
Category: 2-Dimensions
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Sawtoothxy Function

I. Mathematical Expression:

$f(X)=g(r) \cdot h(t)$

where:

$\bullet$ $g(r)=\left[\sin(r)-\frac{\sin(2r)}{2}+\frac{\sin(3r)}{3}-\frac{\sin(4r)}{4}+4\right]\left(\frac{r^2}{r+1}\right)$

$\bullet$ $h(t)=\frac{1}{2}\cos\left(2t-\frac{1}{2}\right)+\cos(t)+2$

$\bullet$ $r=\sqrt{x^2_1+x^2_2}$

$\bullet$ $t=\text{atan2}\left(x_2,x_1\right)$

$\bullet$ $-20\leq x_i\leq 20$ , $i=1,2$

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

$\bullet$ $x^*_i=0$

II. 2&3D-Plots:

III. 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

IV. MATLAB M-File:

% Sawtoothxy Function
% Range of initial points: -20 <= xj <= 20 , j=1,2
% Global minima: (x1,x2)=(0,0)
% f(x1,x2)=0
% Coded by: Ali R. Alroomi | Last Update: 24 March 2015 | www.al-roomi.org

clear
clc
warning off

x1min=-20;
x1max=20;
x2min=-20;
x2max=20;
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)
        [t r] = cart2pol(x1(j),x2(i)); % change to polar coordinates
        h = cos(2*t - 1/2)/2 + cos(t) + 2;
        g = (sin(r) - sin(2*r)/2 + sin(3*r)/3 - sin(4*r)/4 + 4)*r.^2./(r+1);
        f(i) = g.*h;
    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

V. References: