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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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