Convergence of the Proximal Point Method for Metrically Regular Mappings
Francisco J. Aragón Artacho, Asen L. Dontchev, Michel Geoffroy · ESAIM Proceedings · 2007
In this paper we consider the following general version of the proximal point algorithm for solving the inclusion , where T is a set-valued mapping acting from a Banach space X to a Banach space Y. First, choose any sequence of functions with 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 for 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 ( being a solution to ) such that for each initial point one can find a sequence xn generated by the algorithm which is linearly convergent to . Moreover, if the functions gn have their Lipschitz constants convergent to zero, then there exists a sequence starting from which is superlinearly convergent to . Similar convergence results are obtained for the cases when the mapping T is strongly subregular and strongly regular.