Question

On Matlab use BFGS Method to find the minimum of the following function: f(x) = x₁³ - 2x₂x₁² + x₁² - x₁using initial point (x₀, y₀) = (1, 2)^T to start, and stop when f changes less than 0.0001

Answer 1

clear all;
clc;

H=eye(2,2);
syms a;
f=@(x,y) x^3-2*y*x^2+x^2-x;
g=@(x,y) deal(3*x^2-4*x*y+2*x-1,-2*x^2);
% f=@(x,y) 100*(y-x.^2).^2+(1-x).^2;
% g = @(x,y) deal( 400*x.^3-400*x*y+2*x-2,200*y-200*x.^2);
X=[1 2 ]';
N=7;
sum=X;
n=nargin(f);

for i=1:N
[u v]=g(X(1),X(2));
grad=[u v]';
d=-H*grad;
sum=sum+a*d;
z=f(sum(1),sum(2));
fd=matlabFunction(z);
derivative= matlabFunction(diff(fd(a)));

derivative1=matlabFunction(diff(derivative(a)));

my_val = 1;
n = 10;
for i=1:n

my_val=my_val-(derivative(my_val)/subs(sym(derivative1),{a},{my_val}));
end
alpha=my_val;

X=X+alpha*d;
sum=X;
deltax=alpha*d;
  
[n m]=g(X(1),X(2));
grad1=[n m]';
deltag=grad1-grad;
H=H+(deltax-H*deltag)*(deltax-H*deltag)'/(deltag'*(deltax-H*deltag));
%H=H+(deltax*deltax')/(deltax'*deltag)-(H*deltag)*(H*deltag)'/(deltag'*H*deltag);
% d=-H*grad1;

if (abs(n)< 1e-4&&abs(m)<1e-4)
break
end

end
% disp('The Solution is (X,Y):')

disp(vpa(X))