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Convergence of the proximal point method for metrically regular mappings

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.

    Item Type: Article
    Subjects: 49-xx Calculus of variations and optimal control; optimization > 49Jxx Existence theories
    49-xx Calculus of variations and optimal control; optimization > 49Kxx Necessary conditions and sufficient conditions for optimality
    49-xx Calculus of variations and optimal control; optimization > 49Mxx Methods of successive approximations
    65-xx Numerical analysis > 65Kxx Mathematical programming, optimization and variational techniques
    90-xx Economics, operations research, programming, games > 90Cxx Mathematical programming
    Faculty: UNSPECIFIED
    Depositing User: Dr. Francisco Javier Aragón Artacho
    Date Deposited: 06 Sep 2011 15:44
    Last Modified: 06 Sep 2011 15:44
    URI: https://docserver.carma.newcastle.edu.au/id/eprint/915

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