From: Ronald Bruck <bruck@math.usc.edu>
Subject: Re: [Q]: minimum of a convex function on a convex set of a banach space
Date: 17 Sep 2000 14:00:08 -0500
Newsgroups: sci.math.research
Summary: [missing]

In article <8q0t95$5g9@mcmail.cis.McMaster.CA>, 
kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) wrote:

:In article <8pra8c$198d@edrn.newsguy.com>,
:Humberto Jose Bortolossi  <hjbortol@mat.puc-rio.br> wrote:
::Greetings!
::
::Let U a convex set of a banach space B and f: U -> R a convex
::function defined on U.  I would like to know what conditions to
::impose on U (closed? bounded?) in order to make sure that f has a
::minimum on U.
::
::Any references?
:
:A classic:
:David G. Luenberger: Optimization by Vector Space Methods
:John Wiley & Sons 1969
:ISBN 471-55359x
:
:Hope it helps, ZVK(Slavek).

I don't remember off the top of my head, but does Luenberger discuss 
GENERAL convex functions in infinite-dimensional Banach spaces?

Anyway, the usual conditions:

  U non-empty, convex, and weakly compact;

  f proper (i.e. not identically +\infty), convex, 
  lower semi-continuous; 

==> f assumes a minimum.

To get weak compactness one usually works in a REFLEXIVE Banach space, 
where U closed and convex (and nonempty) suffices.  You can allow U to 
be unbounded if you put a growth condition on f (e.g. f(x) --> +\infty 
as ||x||--> +\infty), but these are just variants of the bounded case.

And, of course, if x0 is a minimizer of f then 0 \in \partial f(x0), the 
subdifferential of f at x0.

I would suggest getting a copy of Moreau's lecture notes.

--Ron Bruck

-- 
Due to University fiscal constraints, .sigs may not be exceed one
line.
==============================================================================

From: israel@math.ubc.ca (Robert Israel)
Subject: Re: [Q]: minimum of a convex function on a convex set of a banach space
Date: 17 Sep 2000 14:00:02 -0500
Newsgroups: sci.math.research

In article <8pra8c$198d@edrn.newsguy.com>,
Humberto Jose Bortolossi  <hjbortol@mat.puc-rio.br> wrote:

>Let U a convex set of a banach space B and f: U -> R a convex
>function defined
>on U.  I would like to know what conditions to impose on U
>(closed? bounded?) in
>order to make sure that f has a minimum on U.

It's not true in general, even if f is linear and U is 
a closed ball.  I think you'll want U to be compact in some
topology in which f is lower semicontinuous.

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2