[Octave-bug-tracker] [bug #57113] fminunc return wrong Hessian matrix

[Giuseppe]

Follow-up Comment #6, bug #57113 (project octave):

Ok, but note that using welder_mead_min from optim package to get the starting
point for fminunc, this return the correct parameters but a diagonal matrix
for their variance!! Look at this

par =

   2.0179e+07   2.5534e+06

V =

Diagonal Matrix

   1.0000e+00            0
            0   1.0000e+00


I can assure that V, that is the inverse of fisher matrix, is totally
incorrect!


[comment #4 comment #4:]
> For minimizing a quadratic form like @(x) sum (x.^2), BFGS [1] converges in
one iteration from any starting point, but the approximate Hessian is
generally not the right value when it's initially set to the identity (which
is the case in fminunc). This is expected. If you need the Hessian for
something like estimating confidence intervals, you should compute it at the
minimum in some other way.
> 
> We could explore initializing the Hessian more intelligently in fminunc
using, e.g., the ideas here [2]. These would give the right answer using BFGS
for this particular problem, and could conceivably be what Matlab does.
fminunc in Octave actually combines BFGS with a double dogleg trust region
minimization (which is different from Matlab's method, based on their
documentation). Although unfortunately there's no reference provided in the
fminunc code to the algorithm used, there are some details in an old mailing
list thread [3]. But in general, you shouldn't count on the approximate
Hessian from this sort of solver to be accurate.
> 
> [1]
https://en.wikipedia.org/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm
> [2] https://scicomp.stackexchange.com/q/11324
> [3]
https://octave.1599824.n4.nabble.com/algorithm-for-fminunc-td4648956.html

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