On 02/03/2012 11:06 AM, Dan Zuras Intervals wrote:
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.
This is the standard constrained global optimization problem. It is solved either by heuristic methods (unreliable but can be used without knowing much about the function) or (more reliably, but the search may take too long) by branch and
bound methods (in which case interval arithmetic plays a major role).
The branch and bound solver BARON won in 2006 the prestigious
Beale-Orchard-Hays prize; see
http://www.mathprog.org/?nav=boh_2006
The price citation says among others:
''BARON also incorporates techniques from automatic differentiation, interval arithmetic, and other areas to yield an automatic, modular, and relatively efficient solver for the very difficult area of global optimization.''
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.
One needs both, and even the Hessian, for an efficient solution process.
The reason I ask is that it occurs to me that a typical
trial minimum would be in the interior of its bounding
box&
This is not true. The minimum is often at the boundary of the box.