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Global Convergence of a Closed-Loop Regularized Newton Method for Solving Monotone Inclusions in Hilbert Spaces

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Abstract

We analyze the global convergence properties of some variants of regularized continuous Newton methods for convex optimization and monotone inclusions in Hilbert spaces. The regularization term is of Levenberg–Marquardt type and acts in an open-loop or closed-loop form. In the open-loop case the regularization term may be of bounded variation.

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Notes

  1. The compact imbedding of BV([0,T]), the space of real-valued functions of bounded variation on [0,T] equipped with the topology of the intermediate convergence, into L 1(0,T) (see e.g. [10, Theorem 10.1.4], [11, 12]) yields a subsequence of λ n converging almost everywhere on [0,T].

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Acknowledgements

H. Attouch and P. Redont are partially supported by ANR-08-BLAN-0294-03; B.F. Svaiter is partially supported by CNPq grants 302962/2011-5, 474944/2010-7, FAPERJ grant E-26/102.940/2011 and PRONEX-Optimization.

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Authors and Affiliations

  1. I3M UMR CNRS 5149, Université Montpellier II, Pl. E. Bataillon, 34095, Montpellier, France

    H. Attouch & P. Redont

  2. IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil

    B. F. Svaiter

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  1. H. Attouch
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  2. P. Redont
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  3. B. F. Svaiter
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Correspondence to H. Attouch.

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Communicated by Enrique Zuazua.

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Attouch, H., Redont, P. & Svaiter, B.F. Global Convergence of a Closed-Loop Regularized Newton Method for Solving Monotone Inclusions in Hilbert Spaces. J Optim Theory Appl 157, 624–650 (2013). https://doi.org/10.1007/s10957-012-0222-3

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