MATLAB: Newton’s method for minimisation returns a critical point

MATLABminimanewtonroot finding

I am trying to implement the newton's method to find the minima in the Himmelblau function.
The code does work most of the time, but on cases like this where my initial guess is (0.5 , 1) it returns a critical point of the function. I understand this is because the gradient becomes 0 and no new points are generated.
Now my question would be, is this normal with this method? Is there a way of getting around this problem?
Thanks for any help
close all; clear; clc
% Initialisation of variables to use
x0 = [0.5;1];
tol = 1e-4;
maxits = 50;
% Himmelblau function
him = @(x,y) (x.^2 + y - 11).^2 + (x + y.^2 - 7).^2;
% Gradient of the Himmelblau
grad_him = @(x,y) [[4*x.^3 + 4*x.*y - 42*x + 2*y.^2 - 14];[4*y.^3 + 4*x.*y - 26*y + 2*x.^2 - 22]];
% Hessian matrix of the Himmelblau
hessian_him = @(x,y) [[ 12*x.^2 + 4*y - 42 , 4*x + 4*y ];[ 4*x + 4*y , 12*y.^2 + 4*x - 26 ]];
% Call to newton's function and displaying our results accordingly
[r, iters, flag] = newton_min(grad_him,hessian_him,x0,tol,maxits);
fprintf ("<strong>Newton's method</strong>\n\n");
switch (flag)
case 0
fprintf ("There was a convergence on f\n\n");
fprintf("The minima found is: \n");
disp(r);
fprintf("It took %d iterations.\n\n",iters);
case 1
fprintf ("There was a convergence on x\n\n");
fprintf("The minima found is: \n");
disp(r);
fprintf("It took %d iterations.\n\n",iters);
otherwise
fprintf ("There was no convergence\n\n");
end
function [r, iters, flag] = newton_min(dg,ddg,x0,tol,maxits)
x = x0(1); y = x0(2);
r = NaN;
flag = -1;
for iters = 1 : maxits
x_old = [x;y];
x_new = x_old - (ddg(x,y)\dg(x,y));
if norm(dg(x,y)) < tol
flag = 0;
r = x_new;
return;
end
if norm(x_new - x_old) <= (tol + eps*norm(x_new))
flag = 1;
r = x_new;
return;
end
x = x_new(1);
y = x_new(2);
end
end

Best Answer

  • Yes, it's normal.