Issue |
ESAIM: Proc.
Volume 17, 2007
CSVAA 2004 - Control Set-Valued Analysis and Applications
|
|
---|---|---|
Page(s) | 1 - 8 | |
DOI | https://doi.org/10.1051/proc:071701 | |
Published online | 26 April 2007 |
Convergence of the Proximal Point Method for Metrically Regular Mappings
1
Department of Statistics and Operations Research, University of Alicante, 03071 Alicante, Spain, This author is supported by Grant BES-2003-0188 from FPI Program of MEC (Spain).
2
Mathematical Reviews, Ann Arbor, MI 48107, USA, .
3
Laboratoire AOC, Dpt. de Mathématiques, Université Antilles-Guyane, F-97159 Pointe-à-Pitre, Guadeloupe, . This author is supported by Contract EA3591 (France).
In this paper we consider the following general version of
the proximal point algorithm for solving the inclusion , where T is a set-valued mapping acting from a Banach space
X to a Banach space Y. First, choose any sequence of
functions
with
that are Lipschitz
continuous
in a neighborhood of the origin. Then pick
an initial guess x0 and find a sequence xn by applying the
iteration
for
We prove that if the Lipschitz constants of gn are bounded by
half the reciprocal of the modulus of regularity of T, then
there exists a neighborhood O of
(
being a solution
to
) such that for each initial point
one
can find a sequence xn generated by the algorithm which is
linearly convergent to
. Moreover, if the functions gn
have their Lipschitz constants
convergent to zero, then there exists
a sequence starting from
which is superlinearly
convergent to
. Similar convergence results are obtained for
the cases when the mapping T is strongly subregular and strongly
regular.
Résumé
Nous considérons dans ces travaux une généralisation de
l'algorithme du point proximal pour résoudre des inclusions du
type , où T est une application multivoque
agissant entre deux espaces de Banach X et Y. Tout d'abord, on
se donne une suite de fonctions
telle que chaque
fonction gn soit lipschitzienne dans un voisinage de l'origine
et vérifie
. Partant d'un point initial x0 on
construit une suite xn en appliquant la procédure itérative
for
On
montre que si les constantes de lipschitz des fonctions gn sont
bornées supérieurement par la moitié de l'inverse du module
de régularité de T, alors il existe un voisinage O de
(
étant une solution de
) tel que pour
tout point initial
on peut trouver une suite xn
engendrée par l'algorithme et qui converge linéairement vers
. De plus, si la suite des constantes de lipschitz des
fonctions gn converge vers 0, alors il existe une suite
partant du point
et qui converge super-linéairement
vers
. Des résultats de convergence similaires sont obtenus
quand l'application multivoque T est fortement
sous-régulière et fortement régulière.
Mathematics Subject Classification: 49J53 / 49J40 / 90C48
Key words: proximal point algorithm / set-valued mapping / metric regularity / subregularity / strong regularity / variational inequality / optimization.
© EDP Sciences, ESAIM, 2007
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