[SciPy-user] fmin using spherical bounds

[Anne Archibald]

2009/5/21 ElMickerino <elmickerino at hotmail.com>:
>
> 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,
> Michael

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.


Anne

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