A strongly convergent hybrid proximal method in Banach spaces |
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Authors: | Rolando Gá rciga Otero,B.F. Svaiter |
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Affiliation: | a Instituto de Economia da Universidade Federal de Rio de Janeiro, Avenida Pasteur 250, Rio de Janeiro, RJ, 22290-240, Brazil b Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ, 22460-320, Brazil |
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Abstract: | This paper is devoted to the study of strong convergence in inexact proximal like methods for finding zeroes of maximal monotone operators in Banach spaces. Convergence properties of proximal point methods in Banach spaces can be summarized as follows: if the operator have zeroes then the sequence of iterates is bounded and all its weak accumulation points are solutions. Whether or not the whole sequence converges weakly to a solution and which is the relation of the weak limit with the initial iterate are key questions. We present a hybrid proximal Bregman projection method, allowing for inexact solutions of the proximal subproblems, that guarantees strong convergence of the sequence to the closest solution, in the sense of the Bregman distance, to the initial iterate. |
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Keywords: | Proximal point method Relative error Inexact solutions Hybrid steps Strong convergence Enlargement of maximal monotone operators |
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