🚀 go-pugleaf

In article <1993Jan8.140544.25242@noao.edu> nroddier@noao.edu (Nicolas Roddier) writes:
>	I'm looking for some theorems that would garantee that an
>	optimization is an absolute minimum. E-L methods give some
>	necessary conditions for an optimzation to be an extremum.
>	How about some sufficient conditions for being the absolute
>	minimum.

I take it by E-L you mean Euler-Lagrange which means your talking
about optimal control problems (optimization usually means optimizing
over a finite set of parameters as opposed to finding an optimal
control which is infinite-dimenstional).

The Hamiliton-Jacobi equation gives a sufficient condition for an
absolute minimum.  However, it usually can't be solved because its a
system of nonlinear, partial differential equations.  Furthermore, it
depends on being able to compute an absolute minimum of another
function.  As soon as you try to use calculus to compute the minimum
of this other function (i.e.  setting the derivative to zero) you
automatically sacrifice the ability to find the "absolute" or global
minimum.  The only way around this is to start with a convex problem
(like Linear-Quadratic regulator).

If you're willing to settle for a local minimum, there are strengthened
versions of the Pontryagin maximum principle (like E-L) which provide
sufficient conditions for a local minimum.

-Adam Schwartz
adams@robotics.berkeley.edu