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An abstract proximal point algorithm

Author

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  • Laurenţiu Leuştean

    (University of Bucharest
    University of Bucharest
    Simion Stoilow Institute of Mathematics of the Romanian Academy)

  • Adriana Nicolae

    (University of Seville
    Babeş-Bolyai University)

  • Andrei Sipoş

    (Simion Stoilow Institute of Mathematics of the Romanian Academy
    Technische Universität Darmstadt)

Abstract

The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators, fixed points of nonexpansive mappings, as well as minimizing convex functions. The algorithm works by applying successively so-called “resolvent” mappings associated to the original object that one aims to optimize. In this paper we abstract from the corresponding resolvents employed in these problems the natural notion of jointly firmly nonexpansive families of mappings. This leads to a streamlined method of proving weak convergence of this class of algorithms in the context of complete CAT(0) spaces (and hence also in Hilbert spaces). In addition, we consider the notion of uniform firm nonexpansivity in order to similarly provide a unified presentation of a case where the algorithm converges strongly. Methods which stem from proof mining, an applied subfield of logic, yield in this situation computable and low-complexity rates of convergence.

Suggested Citation

  • Laurenţiu Leuştean & Adriana Nicolae & Andrei Sipoş, 2018. "An abstract proximal point algorithm," Journal of Global Optimization, Springer, vol. 72(3), pages 553-577, November.
    • Handle: RePEc:spr:jglopt:v:72:y:2018:i:3:d:10.1007_s10898-018-0655-9
    DOI: 10.1007/s10898-018-0655-9
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    References listed on IDEAS

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    1. L. C. Ceng & B. S. Mordukhovich & J. C. Yao, 2010. "Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 267-303, August.
    2. David Ariza-Ruiz & Genaro López-Acedo & Adriana Nicolae, 2015. "The Asymptotic Behavior of the Composition of Firmly Nonexpansive Mappings," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 409-429, November.
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    Cited by:

    1. Laurenţiu Leuştean & Pedro Pinto, 2021. "Quantitative results on a Halpern-type proximal point algorithm," Computational Optimization and Applications, Springer, vol. 79(1), pages 101-125, May.
    2. Andrei Sipoş, 2022. "Abstract strongly convergent variants of the proximal point algorithm," Computational Optimization and Applications, Springer, vol. 83(1), pages 349-380, September.

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