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An abstract proximal point algorithm

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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.

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Acknowledgements

Adriana Nicolae was partially supported by DGES (MTM2015-65242-C2-1-P). She would also like to acknowledge the Juan de la Cierva - Incorporación Fellowship Program of the Spanish Ministry of Economy and Competitiveness. Laurenţiu Leuştean and Andrei Sipoş were partially supported by a grant of the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, project number PN-II-ID-PCE-2011-3-0383.

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Authors and Affiliations

  1. The Research Institute of the University of Bucharest (ICUB), University of Bucharest, M. Kogălniceanu 36-46, 050107, Bucharest, Romania

    Laurenţiu Leuştean

  2. Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, 010014, Bucharest, Romania

    Laurenţiu Leuştean

  3. Simion Stoilow Institute of Mathematics of the Romanian Academy, Calea Griviţei 21, 010702, Bucharest, Romania

    Laurenţiu Leuştean & Andrei Sipoş

  4. Department of Mathematical Analysis - IMUS, University of Seville, C/ Tarfia s/n, 41012, Seville, Spain

    Adriana Nicolae

  5. Department of Mathematics, Babeş-Bolyai University, Kogălniceanu 1, 400084, Cluj-Napoca, Romania

    Adriana Nicolae

  6. Department of Mathematics, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289, Darmstadt, Germany

    Andrei Sipoş

Authors
  1. Laurenţiu Leuştean
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  2. Adriana Nicolae
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  3. Andrei Sipoş
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Correspondence to Laurenţiu Leuştean.

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Leuştean, L., Nicolae, A. & Sipoş, A. An abstract proximal point algorithm. J Glob Optim 72, 553–577 (2018). https://doi.org/10.1007/s10898-018-0655-9

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  • DOI: https://doi.org/10.1007/s10898-018-0655-9

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