% Example of the lagrange function for a 1-d constrained minimization problem
% f(x)=x^2 --> min subject to constraint x=1
% this problem is trivially simple: the function needs to be minimized just on a single point
% the solution is, of course, minimum at x=1
% the Lagrange's function for this problem looks like 
% L(x,lambda)=x^2+lambda*(x-1) 
% and Lagrange's methods consists in minimizing L(x, lambda) over space of TWO variables x and lambda
% let us check if Lagrange's method still gives the right answer x=1 (value of lambda in unimportant) 
figure(1);
n=20;
% note: y plays the role of lambda
[x,y] = meshgrid([-3:1/n:3],[-5:1/n:5]);
z = x.^2+y.*(x-1);
surf(x,y,z, 'FaceColor', 'green', 'EdgeColor', 'none');
xlabel('original variable');
ylabel('artificial variable');
camlight left;
%%% add a line to indicate where the critical point occure
hold;
[x,y] = meshgrid(1,[-5:1/n:5]);
plot3(x,y,x.^2+y.*(x-1),'r-');
[x,y] = meshgrid([-3:1/n:3],-2);
plot3(x,y,x.^2+y.*(x-1),'b-');