Hi all,
I've been trying to minimize a likelihood function which I have to
evaluate numerically and so there is always a bit of roundoff. I've
been testing several of the algorithms in nlopt and so far the BOBYQA
algorithm seems to do the best, however I've run into one difficulty
that I was wondering if anyone on this list might have dealt with
before and so might be able to provide some guidance.
The basic problem I have is that since the likelihood function is
evaluated numerically it can often happen that the region over which
the BOBYQA algorithm samples becomes too small and so it thinks it has
found a local minimum even though the function isn't changing only
because there is no difference due to small rounding errors. I have
*partially* mitigated this by using a step size which I know is
roughly equal to the width of the minimum and repeatedly running the
minimization like so:
do {
*fmin = fval;
nlopt_optimize(opt,x,&fval)
} while (fval < fmin && (fabs(fval-fmin) > 1e-2));
which will repeatedly run the minimization until the final convergence
point is stable. This has the effect of widening the region in which
the minimizer samples anytime it gets "stuck" which will often allow
it to keep progressing towards the true minimum.
The only problem with this approach is that it seems to be very
inefficient. It is usually necessary for the fit to run in this loop
many times to "kick" the algorithm from some local point where it
thinks it has converged which then requires another 2*n evaluations
for the next fit to get started (my likelihood function takes a really
long time to evaluate, it's on the order of seconds per function
call).
I was thinking that the fundamental issue seems to be that there is a
minimum resolution to the likelihood below which the function calls at
various points are not very reliable estimates when trying to
approximate the Hessian. I know that the fitter Minuit has an option
to set the machine precision size which I've used in the past to tell
it that it should disregard differences in the likelihood below some
value. Alternatively if I could set something like a minimum step size
in each direction to make sure that it didn't try to contract too much
that might also work.
Has anyone else run into this issue or have any ideas on the best way
to solve it?
Thanks in advance for any help and for the great software!
Tony
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