[SciPy-user] constrained optimization
Ondrej Certik
ondrej@certik...
Mon Apr 28 15:03:25 CDT 2008
On Mon, Apr 28, 2008 at 8:57 PM, Robert Kern <robert.kern@gmail.com> wrote:
> On Mon, Apr 28, 2008 at 1:34 PM, John Hunter <jdh2358@gmail.com> wrote:
> > I need to do a N dimensional constrained optimization over a weight w
> > vector with the constraints:
> >
> > * w[i] >=0
> >
> > * w.sum() == 1.0
> >
> > Scanning through the scipy.optimize docs, I see a number of examples
> > where parameters can be bounded by a bracketing interval, but none
> > where constraints can be placed on combinations of the parameters, eg
> > the sum of them. One approach I am considering is doing a bracketed
> > [0,1] constrained optimization over N-1 weights (assigning the last
> > weight to be 1-sum others) and modifying my cost function to punish
> > the optimizer when the N-1 input weights sum to more than one.
> >
> > Is there a better approach?
>
> Transform the coordinates to an unconstrained N-1-dimensional space.
> One such transformation is the Aitchison (or "additive log-ratio")
> transform:
>
> y = log(x[:-1] / x[-1])
>
> And to go back:
>
> tmp = hstack([exp(y), 1.0])
> x = tmp / tmp.sum()
>
> Searching for "compositional data analysis" should yield similar
> transformations, but this one should be sufficient for maintaining
> constraints. For doing statistics, the other have better properties.
Wow, that is very clever. Just today I was thinking how to do it and
it didn't occur to me I should read scipy-user. :)
The exp/log transform is clear, but I didn't figure out that in order
to maintain
the norm, I can maintain it in the last element, so it's enough to do:
y = x[:-1]/x[-1]
tmp = hstack([y, 1.0])
x = tmp / tmp.sum()
Very cool, thanks. However, the transform is not one to one, e.g. both
x = [1, 2, 1, 4]
x = [2, 4, 2, 8]
represent the same thing:
y = [0.25, 0.5, 0.25]
and
Ondrej
More information about the SciPy-user
mailing list