Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE PROXIMAL POINT ALGORITHM IN UNIFORMLY CONVEX METRIC SPACES

  • Choi, Byoung Jin (Department of Mathematics Sungkyunkwan University) ;
  • Ji, Un Cig (Department of Mathematics Research Institute of Mathematical Finance Chungbuk National University)
  • Received : 2015.06.16
  • Published : 2016.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

We introduce the proximal point algorithm in a p-uniformly convex metric space. We first introduce the notion of p-resolvent map in a p-uniformly convex metric space as a generalization of the Moreau-Yosida resolvent in a CAT(0)-space, and then we secondly prove the convergence of the proximal point algorithm by the p-resolvent map in a p-uniformly convex metric space.

Keywords

References

  1. M. Bacak, The proximal point algorithm in metric spaces, Israel J. Math. 194 (2013), no. 2, 689-701. https://doi.org/10.1007/s11856-012-0091-3
  2. I. Beg, Inequalities in metric spaces with applications, Topol. Methods Nonlinear Anal. 17 (2001), no. 1, 183-190. https://doi.org/10.12775/TMNA.2001.012
  3. B. J. Choi, J. Heo, and U. C. Ji, Asymptotic property for inductive means in uniformly convex metric spaces, preprint, 2015.
  4. J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), no. 3, 396-414. https://doi.org/10.1090/S0002-9947-1936-1501880-4
  5. C. Conde, Geometric interpolation in p-Schatten class, J. Math. Anal. Appl. 340 (2008), no. 2, 920-931. https://doi.org/10.1016/j.jmaa.2007.09.008
  6. C. Conde, Nonpositive curvature in p-Schatten class, J. Math. Anal. Appl. 356 (2009), no. 2, 664-673. https://doi.org/10.1016/j.jmaa.2009.03.036
  7. T. Foertsch, Ball versus distance convexity of metric spaces, Beitrage Algebra Geom. 45 (2004), no. 2, 481-500.
  8. H. Fukharuddin, A. R. Khan, and Z. Akhtar, Fixed point results for a generalized nonexpansive map in uniformly convex metric spaces, Nonlinear Anal. 75 (2012), no. 13, 4747-4760. https://doi.org/10.1016/j.na.2012.03.025
  9. J. Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Comment. Math. Helv. 70 (1995), no. 4, 659-673. https://doi.org/10.1007/BF02566027
  10. J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects, Lectures in Mathematics ETH Zurich, Birkhauser Verlag, Basel, 1997.
  11. J. Jost, Nonlinear Dirichlet forms, in New Directions in Dirichlet Forms, AMS/IP Stud. Adv. Math. 8, pp. 1-7, American Mathematical Society, Providence, RI, 1998.
  12. J. Jost, Riemannian Geometry and Geometric Analysis, Fifth ed., Universitext, Springer-Verlag, Berlin, 2008.
  13. M. A. Khamsi and A. R. Khan, Inequalities in metric spaces with applications, Nonlinear Anal. 74 (2011), no. 12, 4036-4045. https://doi.org/10.1016/j.na.2011.03.034
  14. K. Kuwae, Jensen's inequality on convex spaces, Calc. Var. Partial Differential Equations 49 (2014), no. 3-4, 1359-1378. https://doi.org/10.1007/s00526-013-0625-5
  15. J.-J. Moreau, Proximite et dualite dans un espace hilbertien, Bull. Soc. Math. France 93 (1965), 273-299.
  16. A. Naor and L. Silberman, Poincare inequalities, embeddings, and wild groups, Compos. Math. 147 (2011), no. 5, 1546-1572. https://doi.org/10.1112/S0010437X11005343
  17. S. Ohta, Convexities of metric spaces, Geom. Dedicata 125 (2007), 225-250. https://doi.org/10.1007/s10711-007-9159-3
  18. S. Ohta and M. Palfia, Discrete-time gradient flows and law of large numbers in Alexandrov spaces, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1591-1610. https://doi.org/10.1007/s00526-015-0837-y
  19. B. Prus and R. Smarzewski, Strongly unique best approximations and centers in uniformly convex spaces, J. Math. Anal. Appl. 121 (1987), no. 1, 10-21. https://doi.org/10.1016/0022-247X(87)90234-4
  20. T. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topol. Methods Nonlinear Anal. 8 (1996), no. 1, 197-203. https://doi.org/10.12775/TMNA.1996.028
  21. K.-T. Sturm, Probability measures on metric spaces of nonpositive curvature, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), 357-390, Contemp. Math. 338, Amer. Math. Soc., Providence, RI, 2003.
  22. W. Takahashi, A convexity in metric space and nonexpansive mappings. I, Kodai Math. Sem. Rep. 22 (1970), 142-149. https://doi.org/10.2996/kmj/1138846111
  23. H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), no. 12, 1127-1138. https://doi.org/10.1016/0362-546X(91)90200-K

Cited by

  1. Proximal-type algorithms for split minimization problem in P-uniformly convex metric spaces pp.1572-9265, 2018, https://doi.org/10.1007/s11075-018-0633-9
  2. Proximal point algorithm involving fixed point of nonexpansive mapping in 𝑝-uniformly convex metric space vol.0, pp.0, 2019, https://doi.org/10.1515/apam-2018-0026