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From: Alessandro on 2 Jun 2010 13:44
Hi guys
I posted something similar some time ago:
This is a long story, I make it short:
I am working in a project where I need to find a matrix defined by a third degree polynomial.
Originally I thought that a good idea can be to find the solution iteratively using a gradient descent technique, so I implemented a golden section line search in matlab (with the code described below).
The algorithm looks powerful (it finds automatically the perfect step), but unfortunately the golden section line search does not avoid being stuck in local minima.
So I moved to a stochastic approach like simulated annealing, now the problem is that with my objective function the default convergence never happens.
So I have to force annealing to stop. BUT WHEN ?
I tried after some iteration but the annealing has the defect that at some point it randomize his trusted region and maybe when I stop it, it is really far from the minima (I am using too many iterations). And I have the same problem with time limit. So when I should stop the minima the search ?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Code using the golden section:

%Initial third degree polynomial
Cest = initial_guess;
normdev=inf ;
stepsize=inf;

%Stopping condition
stopping_condition = 10^(-5) *norm(X*X'/no_samples,'fro');

while abs(normdev*stepsize) > stopping_condition
%Third degree polynomial
dnew = Cest - 1/no_samples*(X*X' - 2/sigma^2 * (Cest*Cest'*Cest-Cest*B'*Cest));
%Find the best stepsize as a minimum using the goldensection line search
stepsize = fminbnd( @(stepsize) step(stepsize,Cest,dnew,X*X',B,sigma,no_samples),-.1,.1);

%Update
Cest = Cest + stepsize*dnew;
normdev = norm(dnew,'fro');
end

function error = step(stepsize,Cest,dnew,XX,B,sigma,no_samples)
Cest = Cest + stepsize*dnew;
error = norm(Cest - 1/no_samples*(XX - 2/sigma^2 * (Cest^3-Cest*B*Cest)),'fro');

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Code using the simulated annealing:
Cest = initial_point;

lb = -20*ones(size(Cest));
ub = 20*ones(size(Cest));

options = saoptimset('MaxIter',300) ;

Cest = simulannealbnd(@(funcobj) myfuncobj(Cest ,X*X',B,sigma,no_samples),Cest,lb,ub, options);


function error = myfuncobj( Cest, XX, B, sigma, no_samples)
error = norm(Cest - 1/no_samples*(XX - 2/sigma^2 * (Cest^3-Cest*B*Cest)),'fro');
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