🚀 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

In article <1993Jan8.140544.25242@noao.edu> nroddier@noao.edu (Nicolas Roddier) writes:
>Hello control people!
>
>       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.
>
>       Thanks for any help.
>
>       Nick.
>
>----------
>nroddier@noao.edu
>
If you try to find minimization in a bounded region, I recommend Branch and 
Bound method. This method has been introduced to calculate robust control 
problems recently. It is believed that the method can be used to solve 
non-convex optimization problems. Here are a few references:
 
R.R.E.de Gaston and M.G.Safonov, 1988, Exact calculation of the multiloop 
  stability margine, IEEE trans. AC-33, pp.156-171
A.Sederis and R.S.Sanches Pena, 1990, Robustness magin calculation with dynamic
  and real parametric uncertainty, IEEE trans AC-35, pp.970-974.
V.Balakrishnan, S.Boyd and S.Balemi, 1991, Branch and bound algorithm for 
  computing the minimum stability degree of parameter dependent linear systems,
  Int. J. Robust and Nonlinear Control. Vol.1, pp295-317.
M.P.Newlin and P.M.Young, 1992, Mixed mu problems and branch and bound 
  techniques, Proceedings of 31st CDC, pp.3175-3180

...and the references therein.
 
W.Wang