can give two answers, well known
Monday, April 9, 2007
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Minimization of convex functions. The
can give two answers, well known
on the minimization of convex functions: The first can give two answers, well known
. We recall the following formulation
(perhaps more practical):
Theorem.
Let (E, II) a reflexive Banach space.
A is a subset of non-empty, convex and firm E;
f: A ----]- \\ infty, \\ infty] is a convex function
, semi-continuous - inferieuremet, non-
-identity equal to \\ infty and such that:
lim f (x) = \\ infty <= k max(rho(x-y),rho(x-Tx),rho(y-Ty)) sur DxD (*)
IXI -> \\ infty
Then f admits a minimum on A. 
As a simple example f (x) = Ix-aI. (A in E) For proof of this result, see
H. Brezis (References 02/02/2007).
deuxieme.Le The following result generalizes a
resulat well known in the real right
:
Theorem.
Let E be a Banach space is a convex quelconque.f
IR.On in E implies that there exists a z in E such that
d_ {+} f (z) = 0.
(d_ {+} f (z). X = lim [f (z + hx)-f (z)] / h,
h -> 0 ^ {+}
with z, x E h IR)
Then f has a minimum at z
for information on d_ {+} f (z), see
RHMartin (References 02/02/2007).
Then, we easily show the inequality
f (x)> = f (z) + d_ {+} f (z). (Xz)
Finally, note that if f is strictly convex, then , f admits a unique minimum
.
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