[SciPy-user] fmin using spherical bounds
Thu May 21 11:38:51 CDT 2009
On Thu, May 21, 2009 at 11:36, <firstname.lastname@example.org> wrote:
> On Thu, May 21, 2009 at 12:27 PM, Anne Archibald
> <email@example.com> wrote:
>> 2009/5/21 ElMickerino <firstname.lastname@example.org>:
>>> Hello Fellow SciPythonistas,
>>> I have a seemingly simple task: minimize a function inside a (hyper)sphere
>>> in parameter space. Unfortunately, I can't seem to make fmin_cobyla do what
>>> I'd like it to do, and after reading some of the old messages posted to this
>>> forum, it seems that fmin_cobyla will actually wander outside of the allowed
>>> regions of parameter space as long as it smells a minimum there (with some
>>> appropriate hand-waving).
>>> The function I'd like to minimize is only defined in this hypersphere (well,
>>> hyperellipsoid, but I do some linear algebra), so ideally I'd use something
>>> like fmin_bounds to strictly limit where the search can occur, but it seems
>>> that fmin_bounds can only handle rectangular bounds. fmin_cobyla seems to
>>> be happy to simply ignore the constraints I give it (and yes, I've got print
>>> statements that make it clear that it is wandering far, far outside of the
>>> allowed region of parameter space). Is there a simple way to use
>>> fmin_bounds with a bound of the form:
>>> x^2 + y^2 + z^2 + .... <= 1.0 ?
>>> or more generally:
>>> transpose(x).M.x <= 1.0 where x is a column vector and M is a
>>> positive definite matrix?
>>> It seems very bizarre that fmin_cobyla is perfectly happy to wander very,
>>> very far outside of where it should be.
>>> Thanks very much,
>> My experience with this sort of thing has been that while constrained
>> optimizers will only report a minimum satisfying the constraints, none
>> of them (that I have used) can work without evaluating the function
>> outside the bounded region. This is obviously a problem if your
>> function doesn't make any sense out there.
>> I have to agree that reparameterizing your function is the way to go.
>> Rectangular constraints are possible. If evaluating the gradient is
>> too hard, just let the minimizer approximate it (though it shouldn't
>> be too hard to come up with a gradient-conversion matrix so that it's
>> a simple matrix multiply). There's no need to rewrite your function at
>> all; you just use a wrapper function that converts coordinates back
>> from spherical to what your function wants.
> Do you know how well these optimization functions would handle
> discontinuities at the boundary? e.g
> def wrapobjectivefn(x):
> if transpose(x).M.x > 1.0:
> return a_large_number
> return realobjectivefn(x)
> I don't know what the appropriate wrapper for the gradient would be,
> maybe also some large vector.
> I'm doing things like this in matlab, but I haven't tried with the
> scipy minimizers yet.
You would probably want some gradient out there to point it back to
the feasible region, at least roughly.
"I have come to believe that the whole world is an enigma, a harmless
enigma that is made terrible by our own mad attempt to interpret it as
though it had an underlying truth."
-- Umberto Eco
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