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Convergence of the proximal point method for metrically regular mappings

By Francisco Javier Aragón Artacho, Asen L. Dontchev and Michel H. Geoffroy

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

Topics: Proximal point algorithm, Set-valued mapping, Metric regularity, Subregularity, Strong regularity, Variational inequality, Optimization, Matemáticas
Publisher: EDP Sciences
Year: 2007
DOI identifier: 10.1051/proc:071701
OAI identifier: oai:rua.ua.es:10045/8057
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