Aragón Artacho, Francisco J. and Dontchev, Asen L. and Geoffroy, Michel H. (2007) *Convergence of the proximal point method for metrically regular mappings.* ESAIM: Proceedings, 17 . pp. 1-8. ISSN 1270-900X

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## Abstract

In this paper we consider the following general version of the proximal point algorithm for solving the inclusion T(x) Э 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) Э 0) such that for each initial point x0 Є 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 Є 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.

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