Gauss–Seidel optimization routines

This is simplest optimization routine. Using this algorithm optimization parameters are changed separately in each step. Only one parameter can be changed in one step while other are helt as constants.

X_k+1=X_k+ΔX_k , k=0,1,2,…

ΔX_k step of parameter X_k. Parameter is changed until function growth is noticed, and then next parameter follows and so on. After cycle with all parameters is completed, then step is changed to half of its value and repeat cycle again. Optimal point searching ends when there is no function increase and last point is held as optimal point.

Lets see how it works with following function:

Its plot:

Using MATLAB script we get results bellow. In each picture start coordinates are different.

Start coordinates. x=150; y=200;

Start coordinates x=50; y=150;

Other example

Start coordinates x=10; y=10;

Start coordinates x=100; y=200;

Third example

Start coordinates x=10; y=10;

Start coordinates x=50; y=200;

Matlab script

close all;

clear all;

clc;

[X,Y] = meshgrid(-100:1:100, -100:1:100);

Z =3*exp(-((X.^2)/78000) -((Y.^2)/20000))-5*exp(-(((X+31).^2)/123) -(((Y+20).^2)/5000));

xx=100;

yy=100;

contour3(Z,20);

hold on;

% figure(2); mesh(Z);

% figure

% plot3(X,Y,Z)

X=70; Y=100;

X1=0;Y1=0;

X0=X;

Y0=Y;

z=160; figure(1); plot(X,Y,’r*’), hold on;

X=X-xx; Y=Y-yy;

TT1 =3*exp(-((X.^2)/78000) -((Y.^2)/20000))-5*exp(-(((X+31).^2)/123) -(((Y+20).^2)/5053));

for i=1:50

z=z/2;

X=X-z;

Z2 =3*exp(-((X.^2)/78000) -((Y.^2)/20000))-5*exp(-(((X+31).^2)/123) -(((Y+20).^2)/5053));

if Z2>=TT1

TT=Z2;

X1=X;

end;

X=X+(2*z);

Z2 =3*exp(-((X.^2)/78000) -((Y.^2)/20000))-5*exp(-(((X+31).^2)/123) -(((Y+20).^2)/5053));

if Z2>=TT1

TT=Z2;

X1=X;

end;

TT1=TT;

X=X1;

X=X+xx; Y=Y+yy;

figure(1); plot3(X,Y,TT,’g*’), hold on;

XX=[X0 X];

YY=[Y0 Y];

line(XX,YY, ‘LineWidth’,1); hold on;

X0=X;

Y0=Y;

X=X-xx; Y=Y-yy;

Y=Y+z;

Z2 = 3*exp(-((X.^2)/78000) -((Y.^2)/20000))-5*exp(-(((X+31).^2)/123) -(((Y+20).^2)/5053));

if Z2>=TT1

TT=Z2;

Y1=Y;

end;

Y=Y-(2*z);

Z2 = 3*exp(-((X.^2)/78000) -((Y.^2)/20000))-5*exp(-(((X+31).^2)/123) -(((Y+20).^2)/5053));

if Z2>=TT1

TT=Z2;

Y1=Y;

end;

TT1=TT;

Y=Y1;

X=X+xx; Y=Y+yy;

figure(1); plot3(X,Y,TT,’*'), hold on;

XX=[X0 X];

YY=[Y0 Y];

line(XX,YY, ‘LineWidth’,3); hold on;

X0=X;

Y0=Y;

X=X-xx; Y=Y-yy;

end

% [X,Y] = meshgrid(-100:1:100, -100:1:100);

% figure

% contour3(Z,70);

figure

[X,Y] = meshgrid(-100:1:100, -100:1:100);

Z =3*exp(-((X.^2)/78000) -((Y.^2)/20000))-5*exp(-(((X+31).^2)/123) -(((Y+20).^2)/5053));

contour3(Z,100);

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