Forcing strong convergence of proximal point iterations in a Hilbert space |
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Authors: | MV Solodov BF Svaiter |
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Institution: | (1) Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botanico, Rio de Janeiro, RJ 22460-320, Brazil. e-mail: solodov@impa.br, benar@impa.br, BR |
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Abstract: | This paper concerns with convergence properties of the classical proximal point algorithm for finding zeroes of maximal monotone
operators in an infinite-dimensional Hilbert space. It is well known that the proximal point algorithm converges weakly to
a solution under very mild assumptions. However, it was shown by Güler 11] that the iterates may fail to converge strongly
in the infinite-dimensional case. We propose a new proximal-type algorithm which does converge strongly, provided the problem
has a solution. Moreover, our algorithm solves proximal point subproblems inexactly, with a constructive stopping criterion
introduced in 31]. Strong convergence is forced by combining proximal point iterations with simple projection steps onto
intersection of two halfspaces containing the solution set. Additional cost of this extra projection step is essentially negligible
since it amounts, at most, to solving a linear system of two equations in two unknowns.
Received January 6, 1998 / Revised version received August 9, 1999?Published online November 30, 1999 |
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Keywords: | : proximal point algorithm – Hilbert spaces – weak convergence – strong convergence Mathematics Subject Classification (1991): 49M45 90C25 90C33 |
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