Advertisement

Two modified proximal point algorithms in geodesic spaces with curvature bounded above

  • Yasunori Kimura
  • Fumiaki Kohsaka
Article

Abstract

We obtain existence and convergence theorems for two variants of the proximal point algorithm involving proper lower semicontinuous convex functions in complete geodesic spaces with curvature bounded above.

Keywords

\(\text {CAT}(1)\) space Convex function Fixed point Geodesic space Minimizer Proximal point algorithm Resolvent 

Mathematics Subject Classification

Primary 52A41 90C25 Secondary 47H10 47J05 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their helpful comments on the original version of this paper. This work was supported by JSPS KAKENHI Grant Numbers 15K05007 and 17K05372.

References

  1. 1.
    Aoyama, K., Kimura, Y., Takahashi, W., Toyoda, M.: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 67, 2350–2360 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aoyama, K., Kohsaka, F., Takahashi, W.: Proximal point methods for monotone operators in Banach spaces. Taiwan. J. Math. 15, 259–281 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ariza-Ruiz, D., Leuştean, L., Lóopez-Acedo, G.: Firmly nonexpansive mappings in classes of geodesic spaces. Trans. Am. Math. Soc. 366, 4299–4322 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bačák, M.: The proximal point algorithm in metric spaces. Isr. J. Math. 194, 689–701 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bačák, M.: Convex Analysis and Optimization in Hadamard Spaces. De Gruyter, Berlin (2014)zbMATHGoogle Scholar
  6. 6.
    Bačák, M., Reich, S.: The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces. J. Fixed Point Theory Appl. 16, 189–202 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bargetz, C., Dymond, M., Reich, S.: Porosity results for sets of strict contractions on geodesic metric spaces. Topol. Methods Nonlinear Anal. 50, 89–124 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Bauschke, H.H., Matoušková, E., Reich, S.: Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal. 56, 715–738 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brézis, H., Lions, P.-L.: Produits infinis de résolvantes. Isr. J. Math. 29, 329–345 (1978)CrossRefzbMATHGoogle Scholar
  11. 11.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  12. 12.
    Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houst. J. Math. 3, 459–470 (1977)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. American Mathematical Society, Providence, RI (2001)CrossRefzbMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Espínola, R., Fernández-León, A.: \({\text{ CAT }}(k)\)-spaces, weak convergence and fixed points. J. Math. Anal. Appl. 353, 410–427 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Espínola, R., Nicolae, A.: Proximal minimization in \({\text{ CAT }}(\kappa )\) spaces. J. Nonlinear Convex Anal. 17, 2329–2338 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker Inc, New York (1984)zbMATHGoogle Scholar
  18. 18.
    Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jost, J.: Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70, 659–673 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kamimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106, 226–240 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kendall, W.S.: Convexity and the hemisphere. J. Lond. Math. Soc. (2) 43, 567–576 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kimura, Y., Kohsaka, F.: Spherical nonspreadingness of resolvents of convex functions in geodesic spaces. J. Fixed Point Theory Appl. 18, 93–115 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kimura, Y., Kohsaka, F.: Two modified proximal point algorithms for convex functions in Hadamard spaces. Linear Nonlinear Anal. 2, 69–86 (2016)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kimura, Y., Kohsaka, F.: The proximal point algorithm in geodesic spaces with curvature bounded above. Linear Nonlinear Anal. 3, 133–148 (2017)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kimura, Y., Saejung, S.: Strong convergence for a common fixed point of two different generalizations of cutter operators. Linear Nonlinear Anal. 1, 53–65 (2015)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Kimura, Y., Satô, K.: Convergence of subsets of a complete geodesic space with curvature bounded above. Nonlinear Anal. 75, 5079–5085 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kimura, Y., Satô, K.: Halpern iteration for strongly quasinonexpansive mappings on a geodesic space with curvature bounded above by one. Fixed Point Theory Appl. 2013(7), 1–14 (2013)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kimura, Y., Saejung, S., Yotkaew, P.: The Mann algorithm in a complete geodesic space with curvature bounded above. Fixed Point Theory Appl. 2013(336), 1–13 (2013)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Kirk, W.A., Panyanak, B.: A concept of convergence in geodesic spaces. Nonlinear Anal. 68, 3689–3696 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lim, T.C.: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 60, 179–182 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Martinet, B.: Régularisation d’inéquations variationnelles par approximations successives. Rev. Française Informat. Recherche Opérationnelle 4, 154–158 (1970)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Mayer, U.F.: Gradient flows on nonpositively curved metric spaces and harmonic maps. Commun. Anal. Geom. 6, 199–253 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Nevanlinna, O., Reich, S.: Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces. Isr. J. Math. 32, 44–58 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ohta, S., Pálfia, M.: Discrete-time gradient flows and law of large numbers in Alexandrov spaces. Calc. Var. Partial Differ. Equ. 54, 1591–1610 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Reich, S., Shafrir, I.: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15, 537–558 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Saejung, S., Yotkaew, P.: Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal. 75, 742–750 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Solodov, M.V., Svaiter, B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 87, 189–202 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Takahashi, W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama (2009)zbMATHGoogle Scholar
  40. 40.
    Xu, H.-K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. (2) 66, 240–256 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Yokota, T.: Convex functions and barycenter on \({\text{ CAT }}(1)\)-spaces of small radii. J. Math. Soc. Jpn. 68, 1297–1323 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Information ScienceToho UniversityMiyama, FunabashiJapan
  2. 2.Department of Mathematical SciencesTokai UniversityKitakaname, HiratsukaJapan

Personalised recommendations