Re: How do you do it...?
On 2/3/2012 5:06 AM, Dan Zuras Intervals wrote:
Hi folks,
A question came up at dinner tonight. And it got me
thinking how an interval guy would solve it. I know
this is off topic to our current discussion but you
are the only interval guys I know. :-)
Suppose I have a potential function p(X) parameterized
by some multi-dimensional X (possibly many dimensions).
The parameters are further constrained by some vector
C(X) = 0 (also possibly many constraints).
The problem is to find the absolute minimum of the
potential subject to the constraints within which
there may be many local minima.
Now I suppose you could solve it by B&B or some
vector form of Newton's. That's not my question.
My question is: Do you work with the potential directly
or do you try to find zeros in its gradient? Actually,
I suppose that would be zeros in some norm on the
gradient.
The reason I ask is that it occurs to me that a typical
trial minimum would be in the interior of its bounding
box& therefore not something that is computed by
looking at the endpoints. And while zeros to the
gradient would also likely be in the interior, one
could instead "score" a bounding box by something like
the minimax of its endpoints.
Both are done in floating-point with good& bad aspects
to each. I'm curious what intervals brings to the table
that might be different in some qualitative fashion.
Anything?
Here are a couple of simple applications from a while back,
involving optimization of potential functions using interval methods.
Y. Lin and M. A. Stadtherr, “Locating Stationary Points of
Sorbate-Zeolite Potential Energy Surfaces Using Interval
Analysis,” J. Chem. Phys., 121, 10159-10166 (2004).
Y. Lin and M. A. Stadtherr, “Deterministic Global Optimization
of Molecular Structures Using Interval Analysis,”
J. Comput. Chem., 26, 1413-1420 (2005).
In the latter, B&B is combined with interval Newton for zeros
of gradient.
I apologise for the digression.
Dan
--
Mark A. Stadtherr
Keating-Crawford Professor
Department of Chemical and Biomolecular Engineering
University of Notre Dame
Notre Dame, IN 46556
Phone: (574) 631-9318
Fax: (574) 631-8366
Email: markst@xxxxxx
WWW: http://www.nd.edu/~markst