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Quantitative results on a Halpern-type proximal point algorithm

Computational Optimization and Applications volume 79, pages 101–125 (2021)Cite this article

Abstract

We apply proof mining methods to analyse a result of Boikanyo and Moroşanu on the strong convergence of a Halpern-type proximal point algorithm. As a consequence, we obtain quantitative versions of this result, providing uniform effective rates of asymptotic regularity and metastability.

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References

  1. Aoyama, K., Toyoda, M.: Approximation of zeros of accretive operators in a Banach space. Israel J. Math. 220, 803–816 (2017)

    Article  MathSciNet  Google Scholar 

  2. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces (2nd edition). Springer, Berlin (2017)

    Book  Google Scholar 

  3. Boikanyo, O.A., Moroşanu, G.: A proximal point algorithm converging strongly for general errors. Optim. Lett. 4, 635–641 (2010)

    Article  MathSciNet  Google Scholar 

  4. Boikanyo, O.A., Moroşanu, G.: Inexact Halpern-type proximal point algorithm. J. Glob. Optim. 51, 11–26 (2011)

    Article  MathSciNet  Google Scholar 

  5. Borwein, J., Reich, S., Shafrir, I.: Krasnoselski-Mann iterations in normed spaces. Can. Math. Bull. 35, 21–28 (1992)

    Article  MathSciNet  Google Scholar 

  6. Boţ, R.I., Csetnek, E.R., Meier, D.: Inducing strong convergence into the asymptotic behaviour of proximal splitting algorithms in Hilbert spaces. Optim. Methods Softw. 34, 489–514 (2019)

    Article  MathSciNet  Google Scholar 

  7. Browder, F.E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Ration. Mech. Anal. 24, 82–90 (1967)

    Article  Google Scholar 

  8. Browder, F.E., Petryshyn, W.V.: The solution by iteration of nonlinear functional equations in Banach spaces. Bull. Am. Math. Soc. 72, 571–575 (1966)

    Article  MathSciNet  Google Scholar 

  9. Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)

    Article  MathSciNet  Google Scholar 

  10. Ferreira, F., Leuştean, L., Pinto, P.: On the removal of weak compactness arguments in proof mining. Adv. Math. 354, 106728 (2019)

    Article  MathSciNet  Google Scholar 

  11. Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)

    Article  MathSciNet  Google Scholar 

  12. Halpern, B.: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957–961 (1967)

    Article  MathSciNet  Google Scholar 

  13. Kamimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106, 226–240 (2000)

    Article  MathSciNet  Google Scholar 

  14. Kohlenbach, U.: Some logical metatheorems with applications in functional analysis. Trans. Am. Math. Soc. 357, 89–128 (2005)

    Article  MathSciNet  Google Scholar 

  15. Kohlenbach, U.: Applied Proof Theory: Proof Interpretations and Their Use in Mathematics. Springer, Berlin (2008)

    MATH  Google Scholar 

  16. Kohlenbach, U.: Proof-theoretic methods in nonlinear analysis. In: Sirakov, B., Ney de Souza, P., Viana, M. (eds.), Proceedings of ICM 2018, Vol. 2, pp. 61–82. World Scientific (2019)

  17. Kohlenbach, U.: On quantitative versions of theorems due to F. E. Browder and R. Wittmann. Adv. Math. 226, 2764–2795 (2011)

    Article  MathSciNet  Google Scholar 

  18. Kohlenbach, U.: Local formalizations in nonlinear analysis and related areas and proof-theoretic tameness. In: Weingartner, P., Leeb, H.-P. (eds.), Kreisel’s Interests. On the Foundations of Logic and Mathematics, pp. 45–61. College Publications (2020)

  19. Kohlenbach, U.: Quantitative analysis of a Halpern-type Proximal Point Algorithm for accretive operators in Banach spaces. J. Nonlinear Convex Anal. 21, 2125–2138 (2020)

    MathSciNet  Google Scholar 

  20. Kohlenbach, U.: Quantitative results on the proximal point algorithm in uniformly convex Banach spaces. J. Convex Anal. 28, 11–18 (2021)

    MathSciNet  MATH  Google Scholar 

  21. Kohlenbach, U., Leuştean, L.: Effective metastability of Halpern iterates in CAT(0) spaces. Adv. Math. 231, 2526–2556 (2012). Addendum in Adv. Math. 250, 650–651 (2014)

  22. Kohlenbach, U., Leuştean, L., Nicolae, A.: Quantitative results on Fejér monotone sequences. Commun. Contemp. Math. 20, 1750015 (2018)

    Article  MathSciNet  Google Scholar 

  23. Kohlenbach, U., Sipoş, A.: The finitary content of sunny nonexpansive retractions. Commun. Contemp. Math. 23, 1950093 (2021)

    Article  MathSciNet  Google Scholar 

  24. Leuştean, L.: Rates of asymptotic regularity for Halpern iterations of nonexpansive mappings. J. Univ. Comput. Sci. 13, 1680–1691 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Leuştean, L., Nicolae, A.: Effective results on nonlinear ergodic averages in \(CAT(\kappa \)) spaces. Ergod. Theory Dyn. Syst. 36, 2580–2601 (2016)

    Article  MathSciNet  Google Scholar 

  26. Leuştean, L., Nicolae, A., Sipoş, A.: An abstract proximal point algorithm. J. Glob. Optim. 72, 553–577 (2018)

    Article  MathSciNet  Google Scholar 

  27. Leuştean, L., Sipoş, A.: An application of proof mining to the proximal point algorithm in CAT(0) spaces. In: Bellow, A., Calude, C., Zamfirescu, T. (eds.) Mathematics Almost Everywhere. In Memory of Solomon Marcus, pp. 153–168. World Scientific (2018)

  28. Leuştean, L., Sipoş, A.: Effective strong convergence of the proximal point algorithm in CAT(0) spaces. J. Nonlinear Variational Anal. 2, 219–228 (2018)

    Article  Google Scholar 

  29. Martinet, B.: Régularisation d’inéquations variationnelles par approximations successives. Revue Française d’Informatique et de Recherche Opérationnelle 4, 154–158 (1970)

    MATH  Google Scholar 

  30. Neumann, E.: Computational problems in metric fixed point theory and their Weihrauch degrees. Log. Methods Comput. Sci. 11, 1–44 (2015)

    Article  MathSciNet  Google Scholar 

  31. Pinto, P.: A rate of metastability for the Halpern type Proximal Point Algorithm. arXiv:1912.12468 [math.FA] (2019)

  32. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  Google Scholar 

  33. Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. Ser. A 87, 189–202 (2000)

    Article  MathSciNet  Google Scholar 

  34. Tao, T.: Soft analysis, hard analysis, and the finite convergence principle (essay posted May 23, 2007). In: Tao,T. Structure and Randomness: Pages from Year One of a Mathematical Blog, pp. 77–87. American Mathematical Society (2008)

  35. Tao, T.: Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Theory Dyn. Syst. 28, 657–688 (2008)

    Article  MathSciNet  Google Scholar 

  36. Walsh, M.: Norm convergence of nilpotent ergodic averages. Ann. Math. 175, 1667–1688 (2012)

    Article  MathSciNet  Google Scholar 

  37. Wang, Y., Wang, F., Xu, H.-K.: Error sensitivity for strongly convergent modifications of the proximal point algorithm. J. Optim. Theory Appl. 168, 901–916 (2016)

    Article  MathSciNet  Google Scholar 

  38. Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik 58, 486–491 (1992)

    Article  MathSciNet  Google Scholar 

  39. Xu, H.-K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Laurenţiu Leuştean was partially supported by a grant of the Romanian Ministry of Research and Innovation, Program 1 - Development of the National RDI System, Subprogram 1.2 - Institutional Performance - Projects for Funding the Excellence in RDI, Contract Number 15PFE/2018. Pedro Pinto acknowledges and is thankful for the financial support of: FCT - Fundação para a Ciência e a Tecnologia under the Project UID/MAT/04561/2019; the research center Centro de Matemática, Aplicações Fundamentais com Investigação Operacional, Universidade de Lisboa; and the ‘Future Talents’ short-term scholarship at Technische Universität Darmstadt. The paper also benefited from discussions with Fernando Ferreira and Ulrich Kohlenbach.

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Authors and Affiliations

  1. Research Center for Logic, Optimization and Security (LOS), Department of Computer Science, Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, 010014, Bucharest, Romania

    Laurenţiu Leuştean

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

    Laurenţiu Leuştean

  3. Department of Mathematics, Technische Universität Darmstadt, Schlossgartenstraße 7, 64289, Darmstadt, Germany

    Pedro Pinto

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  1. Laurenţiu Leuştean
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  2. Pedro Pinto
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Correspondence to Laurenţiu Leuştean.

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Leuştean, L., Pinto, P. Quantitative results on a Halpern-type proximal point algorithm. Comput Optim Appl 79, 101–125 (2021). https://doi.org/10.1007/s10589-021-00263-w

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