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Issue ESAIM: Proc.
Volume 17, 2007
CSVAA 2004 - Control Set-Valued Analysis and Applications
Page(s) 1 - 8
DOI http://dx.doi.org/10.1051/proc:071701
Published online 26 April 2007

ESAIM: Proc., April 2007, Vol. 17, pp. 1-8
DOI: 10.1051/proc:071701

Convergence of the Proximal Point Method for Metrically Regular Mappings

F. J. Aragón Artacho1, A. L. Dontchev2 and M. H. Geoffroy3

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).


(April 29, 2005. / Published online: 26 April 2007)

Abstract
In this paper we consider the following general version of the proximal point algorithm for solving the inclusion $T(x) \ni 0$, where T is a set-valued mapping acting from a Banach space X to a Banach space Y. First, choose any sequence of functions $g_n:X\to Y$ with gn(0) = 0 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 $g_n(x_{n+1}-x_n) +T(x_{n+1}) \ni 0 $ for $n = 0,1,\ldots$ 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 $\overline{x}$ ( $\overline{x}$ being a solution to $T(x) \ni 0$) such that for each initial point $x_0 \in O$ one can find a sequence xn generated by the algorithm which is linearly convergent to $\overline{x}$. Moreover, if the functions gn have their Lipschitz constants convergent to zero, then there exists a sequence starting from $x_0 \in O$ which is superlinearly convergent to $\overline{x}$. 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 $T(x) \ni 0$, 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 $g_n:X\to Y$ telle que chaque fonction gn soit lipschitzienne dans un voisinage de l'origine et vérifie gn(0)=0. Partant d'un point initial x0 on construit une suite xn en appliquant la procédure itérative $g_n(x_{n+1}-x_n) +T(x_{n+1}) \ni 0 $ for $n = 0,1,\ldots$ 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 $\overline{x}$ ( $\overline{x}$ étant une solution de $T(x) \ni 0$) tel que pour tout point initial $x_0 \in O$ on peut trouver une suite xn engendrée par l'algorithme et qui converge linéairement vers $\overline{x}$. De plus, si la suite des constantes de lipschitz des fonctions gn converge vers 0, alors il existe une suite partant du point $x_0 \in O$ et qui converge super-linéairement vers $\overline{x}$. 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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