bfgs.m

% bfgs.m
%
% Quadratic case --- Quasi-Newton BFGS formula 
%
% Step size is determined analytically.
%
%      E(x,y) = ax^2+bxy+cy^2+dx+ey+f
%
% ===> E(x,y) = 1/2 * t'*A*t+B'*t+C  where t = [x; y]'
%
%    E. Mizutani  and  J.-S. Roger Jang,  June 9, 1996
%
a=1;b=1;c=1;d=-1;e=1;f=0;
A = [2*a b; b  2*c];
B = [d; e];
C = f;
point_n = 25;
x=linspace(-3, 3, point_n);
y=linspace(-2, 2, point_n);
[xx,yy]=meshgrid(x,y);
zz=a*xx.^2+b*xx.*yy+c*yy.^2+d*xx+e*yy+f;

subplot(221);
mesh(xx, yy, zz);
view([-20, 25]); set(gca, 'box', 'on');
axis([-inf inf -inf inf -inf inf]);
xlabel('X'); ylabel('Y'); zlabel('E(X,Y)');
title('(a)');

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

subplot(222);
% The initial point
x0 = -3; y0 = 1;     

% How many points ?
p_count = 3;

position = zeros(p_count, 2);
position(1,1) = x0; position(1,2) = y0;
t = position(1, :)';
contour_level = 1/2 * t'*A*t+B'*t+C;

g = A*t+B;
siz = size(A); 
% Set the identity matrix to the initial inverse-Hessian-matrix
% The first step is in the steepest descent direction
InvHess = eye(siz);

for i = 2:p_count,

	dirc = - InvHess * g;
% Determine the step size (eta) analytically.
	eta = - g'* dirc/(dirc'*A*dirc);
	position(i,:)=position(i-1,:) + eta*dirc';

	t = position(i,:)';
	contour_level = [contour_level; 1/2 * t'*A*t+B'*t+C];

	g_old = g;
	g = A*t+B;

	d_pos = position(i,:)'-position(i-1,:)';
	d_g = g - g_old;

	InvHess = ( eye(siz) - ( (d_pos*d_g') / (d_pos'*d_g) ) ) ...
	* InvHess * ( eye(siz) - (d_g*d_pos')/(d_pos'*d_g) ) ...
	+ (d_pos*d_pos')/(d_pos'*d_g);

end

contour(xx, yy, zz, contour_level);
hold on;
% The last point?
x_last = position(p_count,1); y_last = position(p_count,2);
plot(x_last, y_last, 'x');
plot(x0, y0, '*');
arrow(position(:, 1), position(:, 2));
xlabel('X'); ylabel('Y'); title('(b)');
hold off;
axis image;
%