Skip to main content
SpringerLink
Log in

Abstract strongly convergent variants of the proximal point algorithm

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

We prove an abstract form of the strong convergence of the Halpern-type and Tikhonov-type proximal point algorithms in CAT(0) spaces. In addition, we derive uniform and computable rates of metastability (in the sense of Tao) for these iterations using proof mining techniques.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Aleksandrov, A.D.: A theorem on triangles in a metric space and some of its applications. Trudy Math. Inst. Steklov 38, 4–23 (1951)

    MathSciNet  Google Scholar 

  2. Aoyama, K., Toyoda, M.: Approximation of zeros of accretive operators in a Banach space. Israel J. Math. 220(2), 803–816 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ariza-Ruiz, D., Leuştean, L., López-Acedo, G.: Firmly nonexpansive mappings in classes of geodesic spaces. Trans. Amer. Math. Soc. 366, 4299–4322 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ariza-Ruiz, D., López-Acedo, G., Nicolae, A.: The asymptotic behavior of the composition of firmly nonexpansive mappings. J. Optim. Theory Appl. 167, 409–429 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bačák, M.: The proximal point algorithm in metric spaces. Israel J. Math. 194, 689–701 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bačák, M.: Convex analysis and optimization in Hadamard spaces. De Gruyter (2014)

  7. Bauschke,H., Combettes, P.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Second Edition. Springer (2017)

  8. Berg, I.D., Nikolaev, I.G.: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedicata 133, 195–218 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brézis, H., Lions, P.L.: Produits infinis de resolvantes. Israel J. Math. 29, 329–345 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bridson, M., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren der mathematischen wissenschaften, vol. 319. Springer, Berlin (1999)

  11. Browder, F.E.: Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 100, 201–225 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bruck, R.E., Jr.: Nonexpansive projections on subsets of Banach spaces. Pacific J. Math. 47, 341–355 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cheval, H., Kohlenbach, U., Leuştean, L.: On modified Halpern and Tikhonov-Mann iterations. arXiv:2203.11003 [math.OC], (2022)

  14. Cheval, H., Leuştean, L.: Quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration. arXiv:2107.07176 [math.OC], (2021). To appear in: Optim. Methods Softw

  15. Dinis, B., Pinto, P.: On the convergence of algorithms with Tikhonov regularization terms. Optim. Lett. 15(4), 1263–1276 (2021)

    Article  MathSciNet  Google Scholar 

  16. Dinis, B., Pinto, P.: Effective metastability for a method of alternating resolvents. arXiv:2101.12675 [math.FA], (2021). To appear in: Fixed Point Theory

  17. Eckstein, J.: The lions-mercier splitting algorithm and the alternating direction method are instances of the proximal point algorithm. Report LIDS-P-1769, Laboratory for Information and Decision Sciences, MIT (1989)

  18. Goebel, K., Reich, S.: Iterating holomorphic self-mappings of the Hilbert ball. Proc. Japan Acad. Ser. A Math. Sci. 58(8), 349–352 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Goebel, K. Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Monographs and Textbooks in Pure and Applied Mathematics, 83. Marcel Dekker, Inc., New York (1984)

  20. Gromov, M.: Hyperbolic groups. In: S. M. Gersten (ed.), Essays in group theory. Math. Sci. Res. Inst. Publ., 8, pp. 75–264. , Springer, New York (1987)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Minty, G.J.: Fixed points of nonexpanding maps. Proc. Nat. Acad. Sci. USA 50, 1038–1041 (1963)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Kohlenbach, U.: Applied proof theory: Proof interpretations and their use in mathematics. Springer Monographs in Mathematics, Springer (2008)

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Kohlenbach, U.: On the quantitative asymptotic behavior of strongly nonexpansive mappings in Banach and geodesic spaces. Israel J. Math. 216(1), 215–246 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

  29. Kohlenbach, U.: Quantitative analysis of a Halpern-type Proximal Point Algorithm for accretive operators in Banach spaces. J. Nonlinear Convex Anal. 21(9), 2125–2138 (2020)

    MathSciNet  MATH  Google Scholar 

  30. 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)

  31. Kohlenbach, U., López-Acedo, G., Nicolae, A.: Quantitative asymptotic regularity for the composition of two mappings. Optimization 66, 1291–1299 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kohlenbach, U., Pinto, P.: Quantitative translations for viscosity approximation methods in hyperbolic spaces. J. Math. Anal. Appl. 507(2), 125823 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kohlenbach, U., Sipoş, A.: The finitary content of sunny nonexpansive retractions. Commun. Contemp. Math. 23(1), 19550093 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lehdili, N., Moudafi, A.: Combining the proximal algorithm and Tikhonov regularization. Optimization 37(3), 239–252 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  35. Leuştean, L., Nicolae, A.: A note on an alternative iterative method for nonexpansive mappings. J. Convex Anal. 24, 501–503 (2017)

    MathSciNet  MATH  Google Scholar 

  36. Leuştean, L., Nicolae, A., Sipoş, A.: An abstract proximal point algorithm. J. Global Optim. 72(3), 553–577 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. Leuştean, L., Pinto, P.: Quantitative results on a Halpern-type proximal point algorithm. Comput. Optim. Appl. 79(1), 101–125 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lim, T.C.: Remarks on some fixed point theorems. Proc. Amer. Math. Soc. 60, 179–182 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  39. Maingé, P.-E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16(7–8), 899–912 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Martinet, B.: Régularisation d’inéquations variationnelles par approximations successives. Rev. Française Informat. Recherche Opérationnelle 4, 154–158 (1970)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Pinto, P.: A rate of metastability for the Halpern type Proximal Point Algorithm. Numer. Funct. Anal. Optim. 42(3), 320–343 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  43. Reich, S.: Extension problems for accretive sets in Banach spaces. J. Funct. Anal. 26(4), 378–395 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  44. Reich, S., Shafrir, I.: The asymptotic behavior of firmly nonexpansive mappings. Proc. Amer. Math. Soc. 101(2), 246–250 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  45. Reich, S., Shafrir, I.: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15(6), 537–558 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  47. Saejung, S.: Halpern’s iteration in Banach spaces. Nonlinear Anal. 73, 3431–3439 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. Saejung, S.: Halpern’s iteration in CAT(0) spaces. Fixed Point Theory Appl. 2010, 471781 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Suzuki, T.: Reich’s problem concerning Halpern’s convergence. Arch. Math. (Basel) 92, 602–613 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  50. Sipoş, A.: Revisiting jointly firmly nonexpansive families of mappings. arXiv:2006.02167 [math.OC], (2020). To appear in: Optimization

  51. Suzuki, T.: Moudafi’s viscosity approximations with Meir-Keeler contractions. J. Math. Anal. Appl. 325(1), 342–352 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  52. Tao, T.: Soft analysis, hard analysis, and the finite convergence principle. Essay posted May 23, 2007. Appeared in: T. Tao, Structure and Randomness: Pages from Year One of a Mathematical Blog. AMS, 298 pp., 2008

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

    Article  MathSciNet  MATH  Google Scholar 

  54. Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. (Basel) 58, 486–491 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  56. Xu, H.K.: A regularization method for the proximal point algorithm. J. Global Optim. 36, 115–125 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I would like to thank Ulrich Kohlenbach and Laurenţiu Leuştean for their suggestions. This work has been supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI – UEFISCDI, Project Number PN-III-P1-1.1-PD-2019-0396, within PNCDI III.

Author information

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

    Andrei Sipoş

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

    Andrei Sipoş

Authors
  1. Andrei Sipoş
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrei Sipoş.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sipoş, A. Abstract strongly convergent variants of the proximal point algorithm. Comput Optim Appl 83, 349–380 (2022). https://doi.org/10.1007/s10589-022-00397-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-022-00397-5

Keywords

Mathematics Subject Classification

Search

Navigation