[SciPy-user] fmin using spherical bounds

[josef.pktd at gmail.com]

On Thu, May 21, 2009 at 12:27 PM, Anne Archibald
<peridot.faceted at gmail.com> wrote:
> 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
>

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
    else:
          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.

Josef