• To: mathgroup at smc.vnet.net
• Subject: [mg24111] Re: FindMinimum and Gradient
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Wed, 28 Jun 2000 02:11:36 -0400 (EDT)
• Organization: Universitaet Leipzig
• References: <8j9dgm\$512@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Hi,

lets make some high quality computer graphics. If your functions
and the interval [as,bs] boundary look like

\      /
\    /
\__/
+------+
as     bs

the minimum is bracketed. If it looks like

\
\
\
\      /
\    /
\__/

+--+------
as bs

than it is not bracketed. Since this has nothing to do
with the precision it is clear that even with PrecisionGoal->Infinity
FindMinimum[] will not locate a mimimum. You have just to
supply correct input values.

Regards
Jens

Grischa Stegemann wrote:
>
> Dear group
>
> I'm using Mathematica 3.0.2.0 with Solaris.
>
> In trying to minimize relative entropy, i.e. a positive function like
>   S[a_, b_, n_] := -Sum[h[k]*Log[p[k, a, b, n]], {k, -n, n}]
>         /; a > 0 && b > 0
>   S[a_, b_, n_] := -1 /; a <= 0 || b <= 0
> with FindMinimum I just encountered two general problems.
>
> I'm only interested in a,b>0 with a fixed n. Since Mathematica is not able to
> compute the gradient symbolically I specified it with the Gradient option. I
> defined the gradient as 0 if not a,b>0. Thus if FindMinimum gives -1 for the
> minimum i know, that the algorithm went out of the range of interest. But this
> never happened.
>
> Roughly I tried this with several starting points {as,bs}:
> FindMinimum[S[a,b,n0],{a,as},{b,bs},
>   MaxIterations->1000];
>
> First of all, it works fine in many cases (depending on n0 and h). But in the
> other cases I always get
> FindMinimum::"fmlim": "The minimum could not be bracketed in 1000 iterations.
> I tried to play around with Accuracy- and PrecisionGoal but even setting them to
> 1 doesn't make things better.
> So, what exactly does this message mean? I cannot find any explanation of this
> message in the documentation. Where got FindMinimum lost?
>
> The second problem is that due to the first one I wanted to play with the Method
> option.
> But if I try Method->Newton or Method->QuasiNewton Mathematica complains
> FindMinimum::"fmgs":
>     Could not symbolically find the gradient of S[a, b, n]. Try
>     giving two starting values for each variable.
> This is inexplicable since I'm using Gradient->{daS[a,b,n0],dbS[a,b,n0]}, isn't
> it?
>
> --
>                          Grischa Stegemann
>
> ----------------------------------------------------------------------
> Grischa Stegemann                       Technische Universitaet Berlin
> email: Stegemann at physik.tu-berlin.de
>
> *** We are here on Earth to do good for others.
> *** What the others are here for, I do not know.  (W.H. Auden)

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