A multi-step approximant for fixed point problem and convex optimization problem in Hadamard spaces

Abstract

The purpose of this paper is to propose and analyze a multi-step iterative sequence to solve a convex optimization problem and a fixed point problem in an Hadamard space. We aim to establish strong and \( \triangle \)-convergence results of the proposed iterative sequence by employing suitable conditions on the control parameters and the structural properties of the underlying space. As a consequence, we compute an optimal solution for a minimizer of proper convex lower semicontinuous function and a common fixed point of a finite family of total asymptotically quasi-nonexpansive mappings in Hadamard spaces. Our results can be viewed as an extension and generalization of various corresponding results in the existing literature.

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References

  1. 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)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Alber, Ya I., Chidume, C.E., Zegeye, H.: Approximating fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2006, 10673 (2006)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Ambrosio, L., Gigli, N., Savare, G.: Gradient flows in metric spaces and in the space of probability measures. In: 2nd eds., Lectures in Mathematics ETH Zurich. Birkhauser, Basel (2008)

  4. 4.

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

    MathSciNet  Article  Google Scholar 

  5. 5.

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

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bacak, M., Reich, S.: The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces. J. Fixed Point Theory Appl. 16, 189–202 (2014)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bauschke, H.H., Matoušková, E., Reich, S.: Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal. 56, 715–738 (2004)

    MathSciNet  Article  Google Scholar 

  8. 8.

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

    MathSciNet  Article  Google Scholar 

  9. 9.

    Brezis, H., Lions, P.L.: Produits infinis de r ésolvantes. Isr. J. Math. 29, 329–345 (1978)

    Article  Google Scholar 

  10. 10.

    Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. In: Grundelhren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999)

  11. 11.

    Bruck, R.E., Kuczumow, T., Reich, S.: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math. 65, 169–179 (1993)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3, 459–470 (1977)

    MathSciNet  MATH  Google Scholar 

  13. 13.

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

    Article  Google Scholar 

  14. 14.

    Chang, S.S., Cho, Y.J., Zhou, H.: Demiclosed principal and weak convergence problems for asymptotically nonexpansive mappings. J. Korean Math. Soc. 38(6), 1245–1260 (2001)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Chidume, C.E., Ofoedu, E.U.: Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings. J. Math. Anal. Appl. 333, 128–141 (2007)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Cholamjiak, P.: The modified proximal point algorithm in CAT(0) spaces. Optim. Lett. 9, 1401–1410 (2015)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Cholamjiak, P., Abdou, A.A., Cho, Y.J.: Proximal point algorithms involving fixed points of nonexpansive mappings in CAT(0) spaces. Fixed Point Theory Appl. 2015, 227 (2015)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Cholamjiak, W., Cholamjiak, P., Suantai, S.: An inertial forward-backward splitting method for solving inclusion problems in Hilbert spaces. J. Fixed Point Theory Appl. 20, 42 (2018). https://doi.org/10.1007/s11784-018-0526-5

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    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. Springer, New York (2010)

    Google Scholar 

  20. 20.

    Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4, 1168–1200 (2005)

    MathSciNet  Article  Google Scholar 

  21. 21.

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

    MathSciNet  Article  Google Scholar 

  22. 22.

    Fukhar-ud-din, H., Khan, A.R., Khan, M.A.A.: A new implicit algorithm of asymptotically quasi-nonexpansive mappings in uniformly convex Banach spaces. IAENG Int. J. Appl. Math. 42(3), 5 (2012)

    MATH  Google Scholar 

  23. 23.

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

    MathSciNet  Article  Google Scholar 

  24. 24.

    Jost, J.: Convex functionals and generalized harmonicmaps into spaces of non positive curvature. Comment. Math. Helv. 70, 659–673 (1995)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Jost, J.: Nonpositive Curvature: Geometric and Analytic Aspects, Lectures in Mathematics ETH Zurich. Birkhauser Verlag, Basel (1997)

    Google Scholar 

  26. 26.

    Kalsoom, A., Fukhar-ud-din, H., Najib, S.: Proximal point algorithms involving Cesàro type mean of total asymptotically nonexpansive mappings in CAT(0) spaces. Filomat 32, 4165–4176 (2018)

    MathSciNet  Article  Google Scholar 

  27. 27.

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

    MathSciNet  Article  Google Scholar 

  28. 28.

    Kankam, K., Pholasa, N., Cholamjiak, P.: On the convergence and complexity of the modified forward-backward method involving new linesearches for convex minimization. Math. Meth. Appl. Sci., 1-11 (2019)

  29. 29.

    Khan, A.R., Fukhar-ud-din, H., Khan, M.A.A.: An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces. Fixed Point Theory Appl. 2012, 54 (2012). https://doi.org/10.1186/1687-1812-2012-54

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Khan, M.A.A.: Convergence analysis of a multi-step iteration for a finite family of asymptotically quasi-nonexpansive mappings. J. Inequal. Appl. 423, 10 (2013). https://doi.org/10.1186/1029-242X-2013-423

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Khan, M.A.A., Fukhar-ud-din, H.: Convergence analysis of a general iteration schema of nonlinear mappings in hyperbolic spaces. Fixed Point Theory Appl. 238, 18 (2013). https://doi.org/10.1186/1687-1812-2013-238

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Khan, M.A.A., Fukhar-ud-din, H., Kalsoom, A.: Existence and higher arity iteration for total asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces. Fixed Point Theory Appl. 2016, 3 (2016)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Lerkchaiyaphum, K., Phuengrattana, W.: Iterative approaches to solving convex minimization problems and fixed point problems in complete CAT(0) spaces. Numer. Algorithms (2017). https://doi.org/10.1007/s11075-017-0337-6

    Article  MATH  Google Scholar 

  34. 34.

    Li, C., Yao, J.C.: Variational inequalities for set-valued vector fields on Riemannian manifolds: convexity of the solution set and the proximal point algorithm. SIAM J. Control Optim. 50, 2486–2514 (2012)

    MathSciNet  Article  Google Scholar 

  35. 35.

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

    MathSciNet  Article  Google Scholar 

  36. 36.

    Martinet, B.: Regularisation dinéquations variationelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 4, 154–158 (1970)

    MATH  Google Scholar 

  37. 37.

    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)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Pakkaranang, N., Kumam, P., Cho, Y.J.: Proximal point algorithms for solving convex minimization problem and common fixed points problem of asymptotically quasi-nonexpansive mappings in CAT(0) spaces with convergence analysis. Numer. Algorithm 1–19 (2017). https://doi.org/10.1007/s11075-017-0402-1

  39. 39.

    Quiroz, E.A.P.: An extension of the proximal point algorithm with Bregman distances on Hadamard manifolds. J. Glob. Optim. 56, 43–59 (2013)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Reich, S., Salinas, Z.: Weak convergence of infinite products of operators in Hadamard spaces. Rend. Circ. Mat. Palermo 65, 55–71 (2016)

    MathSciNet  Article  Google Scholar 

  41. 41.

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

    MathSciNet  Article  Google Scholar 

  42. 42.

    Smith, S.T.: Optimization techniques on Riemannian manifolds. Fields Inst. Commun. 3, 113–146 (1994)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Suantai, S., Pholasa, N., Cholamjiak, P.: The modified inertial relaxed CQ algorithm for solving the split feasibility problems. J. Ind. Manag. Optim. 14, 1595–1615 (2018)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Sunthrayuth, P., Cholamjiak, P.: Iterative methods for solving quasi-variational inclusion and fixed point problem in q-uniformly smooth Banach spaces. Numer. Algorithms 78, 1019–1044 (2018)

    MathSciNet  Article  Google Scholar 

  45. 45.

    Suparatulatorn, R., Cholamjiak, P., Suantai, S.: On solving the minimization problem and the fixed-point problem for nonexpansive mappings in CAT(0) spaces. Optim. Methods Softw. 32, 182–192 (2017)

    MathSciNet  Article  Google Scholar 

  46. 46.

    Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds, Mathematics and its Applications, vol. 297. Kluwer Academic, Dordrecht (1994)

    Google Scholar 

  47. 47.

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

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The authors are very grateful to the editor and anonymous referees for their helpful comments.

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All authors have contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Muhammad Aqeel Ahmad Khan.

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The author M.A.A. Khan was supported by the Higher Education Commission of Pakistan through project No. NRPU 5332. The author P. Cholamjiak was supported by Thailand Research Fund and University of Phayao under Grant no. RSA 6180084 and Unit of Excellence, University of Phayao.

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Khan, M.A.A., Cholamjiak, P. A multi-step approximant for fixed point problem and convex optimization problem in Hadamard spaces. J. Fixed Point Theory Appl. 22, 62 (2020). https://doi.org/10.1007/s11784-020-00796-3

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Keywords

  • Convex optimization
  • lower semicontinuity
  • proximal point algorithm
  • total asymptotically quasi-nonexpansive mapping
  • common fixed point
  • asymptotic center

Mathematics Subject Classification

  • 47H09
  • 47H10
  • 65K10
  • 65K15