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Newton's method for minimisation returns a critical point

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Dussan Radonich
Dussan Radonich on 10 Nov 2020
Edited: Bruno Luong on 11 Nov 2020
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

Accepted Answer

Matt J
Matt J on 10 Nov 2020
Yes, it's normal.
  30 Comments

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More Answers (2)

J. Alex Lee
J. Alex Lee on 10 Nov 2020
Yes this looks normal, you are only asking to zero the gradient of the function, so naturally that includes non-optimal points where the gradient is [vector] zero.
You can use a non-gradient minimizer, like fminsearch to seek local minima
  1 Comment
Dussan Radonich
Dussan Radonich on 10 Nov 2020
Thank you, the idea is not to used fminsearch as I am trying to compare newton's method against fminsesarch

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Bruno Luong
Bruno Luong on 10 Nov 2020
Edited: Bruno Luong on 10 Nov 2020
"Now my question would be, is this normal with this method?"
Your code juts shows it: yes it is normal.
Now in practice it is very rare that one falls on a stationary point that is not a local minimum. As soon as you work with a non-academic objective function. You won't ever get the gradient == 0 exactly.
"Is there a way of getting around this problem?"
All the book I read about optmization, no one care about this specific problem,since as I said it only happens in academic example. However, many methods will compute for each iteration an approximation of the Hessian, and the positiveness of the Hessian is either enforced or monitored. The Hessian that has negative eigenvalues like yours at (0.5,1) will has automatically a special treatment to escape from non-mimum.

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