Aragón Artacho, Francisco Javier, Dontchev, Asen L., Geoffroy, Michel H.
Convergence of the proximal point method for metrically regular mappings
ARAGÓN ARTACHO, Francisco Javier; DONTCHEV, Asen L.; GEOFFROY, Michel H. "Convergence of the proximal point method for metrically regular mappings". ESAIM: Proceedings. Vol. 17 (Apr. 2007). ISSN 1270-900X, pp. 1-8
URI: http://hdl.handle.net/10045/8057
DOI: 10.1051/proc:071701
ISSN: 1270-900X
Abstract: 
In this paper we consider the following general version of the proximal point algorithm
for solving the inclusion T(x) 3 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 gn : X → 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 gn(xn+1-xn)+T(xn+1) 3 0 for n = 0, 1,... 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 x (x being a solution to T(x) 3 0) such that for each initial
point x0 2 O one can find a sequence xn generated by the algorithm which is linearly convergent
to x. Moreover, if the functions gn have their Lipschitz constants convergent to zero, then there
exists a sequence starting from x0 2 O which is superlinearly convergent to x. Similar convergence
results are obtained for the cases when the mapping T is strongly subregular and strongly regular.
Keywords:Proximal point algorithm, Set-valued mapping, Metric regularity, Subregularity, Strong regularity, Variational inequality, Optimization
EDP Sciences
info:eu-repo/semantics/article