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The modified proximal point algorithm in CAT(0) spaces

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Abstract

We propose a modified proximal point algorithm for solving minimization problems in CAT(0) spaces. We then prove that the sequence converges to a minimizer of convex objective functions. We finally provide the numerical examples for supporting our main result.

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References

  1. Adler, R., Dedieu, J.P., Margulies, J.Y., Martens, M., Shub, M.: Newton’s method on Riemannian manifolds and a geometric model for human spine. IMA J. Numer. Anal. 22, 359–390 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  4. Bačák, M.: Computing medians and means in Hadamard spaces. SIAM J. Optim. 24, 1542–1566 (2014)

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Bridson, M.R.: Haefliger a Metric Spaces of Non-positive Curvature. Grundelhren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999)

    Book  Google Scholar 

  7. Bruhat, M., Tits, J.: Groupes réductifs sur un corps local. I. Données radicielles valuées. Inst. Hautes Études Sci. Publ. Math. 41, 5–251 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  8. Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Bauschke, H.H., Burachik, R., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212 Springer, New York (2011)

  9. Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization. 51, 257–270 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Volume 83 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York. (1984)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  13. Hale, E.T., Yin, W., Zhang, Y.: A fixed-point continuation method for \(\ell _{1}\)-regularized minimization with applications to compressed sensing. Tech. rep., CAAM TR07–07 (2007)

  14. Jost, J.: Convex functionals and generalized harmonic maps into spaces of nonpositive curvature. Comment. Math. Helv. 70, 659–673 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jost, J.: Nonpositive curvature: geometric and analytic aspects. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (1997)

    Book  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Kirk, W.A.: Geodesic geometry and fixed point theory, Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), 195–225. Univ. Sevilla Secr. Publ, Seville (2003)

    Google Scholar 

  18. Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79, 663–683 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Marino, G., Xu, H.K.: Convergence of generalized proximal point algorithm. Comm. Pure Appl. Anal. 3, 791–808 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Mayer, U.F.: Gradient flows on nonpositively curved metric spaces and harmonic maps. Comm. Anal. Geom. 6, 199–253 (1998)

    MATH  MathSciNet  Google Scholar 

  23. Papa Quiroz, E.A., Oliveira, P.R.: Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds. J. Convex Anal. 16, 49–69 (2009)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Saejung, S.: Halpern’s iteration in CAT(0) spaces. Fixed Point Theory Appl. 13 (2010). doi:10.1155/2010/471781

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

    Article  MATH  MathSciNet  Google Scholar 

  27. Shioji, N., Takahashi, W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 125, 3641–3645 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  28. Smith, S.T.: Optimization techniques on Riemannian manifolds. In: Bloch, A. (ed) Fields Institute Communications, vol. 3, pp. 113–146. American Mathematical Society, Providence (1994)

  29. Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)

    MATH  Google Scholar 

  30. Udriste, C.: Convex functions and optimization methods on Riemannian manifolds. Mathematics and its Applications, vol. 297. Kluwer Academic, Dordrecht (1994)

  31. Wang, J.H., Li, C.: Convergence of the family of Euler–Halley type methods on Riemannian manifolds under the \(\gamma \)-condition. Taiwa. J. Math. 13, 585–606 (2009)

    MATH  Google Scholar 

  32. Wang, J.H., López, G.: Modified proximal point algorithms on Hadamard manifolds. Optimization. 60, 697–708 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  35. Yao, Y., Noor, M.A.: On convergence criteria of generalized proximal point algorithms. J. Comput. Appl. Math. 217, 46–55 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B, Stat. Methodol. 67, 301–320 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The author wishes to thank the referees for valuable suggestions. This work was supported by the Higher Education Research Promotion and National Research University project of Thailand, Office of the Higher Education Commission.

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  1. School of Science, University of Phayao, Phayao, 56000, Thailand

    Prasit Cholamjiak

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  1. Prasit Cholamjiak
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Cholamjiak, P. The modified proximal point algorithm in CAT(0) spaces. Optim Lett 9, 1401–1410 (2015). https://doi.org/10.1007/s11590-014-0841-8

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  • DOI: https://doi.org/10.1007/s11590-014-0841-8

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