Skip to main content
SpringerLink
Log in

An extragradient algorithm for solving equilibrium problem and zero point problem in Hadamard spaces

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this article, using a hybrid extragradient method, we introduce a new iterative process for approximating a common element of the set of solutions of an equilibrium problem and a common zero of a finite family of monotone operators in Hadamard spaces. We also give a numerical example to solve a nonconvex optimization problem in an Hadamard space to support our main result.

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

Similar content being viewed by others

References

  1. Bačák, M.: Convex Analysis and Optimization in Hadamard Spaces. De Gruyter Series in Nonlinear Analysis and Applications, vol. 22. De Gruyter, Berlin (2014)

    MATH  Google Scholar 

  2. Berg, I.D., Nikolaev, I.G.: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedi-cata. 133, 195–218 (2008)

    Article  Google Scholar 

  3. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Bridson, M., Haefliger, A.: Metric Spaces of Nonpositive Curvature. Springer, Berlin (1999)

    Book  Google Scholar 

  5. Chang, S., Wang, L., Wang, X.R.: Common solution for a finite family of minimization problem and fixed point problem for a pair of demicontractive mappings in Hadamard spaces. RACSAM 114(2), 12 (2020)

    Article  MathSciNet  Google Scholar 

  6. Chaoha, P., Phon-on, A.: A note on fixed point sets in CAT(0) spaces. J. Math. Anal. Appl. 320, 983–987 (2006)

    Article  MathSciNet  Google Scholar 

  7. Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)

    MathSciNet  MATH  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)

    Chapter  Google Scholar 

  9. Da Cruz Neto, J.X., Ferreira, O.P, Lucambio Pérez, L.R.: Contributions to the study of monotone vector fields. Acta Math. Hungar. 94, 307–320 (2002)

  10. Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Monotone point-to-set vector fields. Balkan J. Geom. Appl. 5, 69–79 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Dehghan, H., Rooin, J.: A characterization of metric projection in CAT(0) spaces. In: International Conference on Functional Equation. Geometric Functions and Applications (ICFGA 2012) 11–12 th May 2012, pp. 41–43. Payame University, Tabriz (2012)

  12. Dhompongsa, S., Kirk, W.A., Sims, B.: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 65, 762–772 (2006)

    Article  MathSciNet  Google Scholar 

  13. Dhompongsa, S., Panyanak, B.: On \(\Delta \)-convergence theorems in CAT(0) spaces. Comput. Math. Appl. 56, 2572–2579 (2008)

    Article  MathSciNet  Google Scholar 

  14. Eskandani, G.Z., Raeisi, M.: On the zero point problem of monotone operators in Hadamard spaces. Numer. Algorithm 80, 1155–1179 (2019)

    Article  MathSciNet  Google Scholar 

  15. Eskandani, G.Z., Raeisi, M., Rassias, T.M.: A hybrid extragradient method for solving pseudomonotone equilibrium problems using Bregman distance. J. Fixed Point Theory Appl. 20, 132 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Heydari, M.T., Khadem, A., Ranjbar, S.: Approximating a common zero of finite family of monotone operators in Hadamard spaces. Optimization. 66, 2233–2244 (2017)

    Article  MathSciNet  Google Scholar 

  18. Iusem, A.N., Mohebbi, V.: Convergence analysis of the extragradient method for equilibrium problems in Hadamard spaces. Comput. Appl. Math. 39, 44 (2020)

    Article  MathSciNet  Google Scholar 

  19. Kakavandi, B.A., Amini, A.: Duality and subdifferential for convex functions on complete CAT(0) metric spaces. Nonlinear Anal. 73, 3450–3455 (2010)

    Article  MathSciNet  Google Scholar 

  20. Khatibzadeh, H., Mohebbi, V.: Approximating solutions of equilibrium problems in Hadamard spaces. Miskolc Math. Notes. 20, 281–297 (2019)

    Article  MathSciNet  Google Scholar 

  21. Khatibzadeh, H., Mohebbi, V.: Monotone and pseudo-monotone equilibrium problems in Hadamard spaces. J. Aust. Math. Soc. (2019). https://doi.org/10.1017/S1446788719000041

  22. Khatibzadeh, H., Ranjbar, S.: Monotone operators and the proximal point algorithm in complete CAT(0) metric spaces. J. Aust. Math. Soc. 103, 70–90 (2017)

    Article  MathSciNet  Google Scholar 

  23. Kirk, W.A., Panyanak, B.: A concept of convergence in geodesic spaces. Nonlinear Anal. 68, 3689–3696 (2008)

    Article  MathSciNet  Google Scholar 

  24. Kumam, P., Chaipunya, P.: Equilibrium problems and proximal algorithms in Hadamard spaces. Optimization. 8, 155–172 (2017)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Martinet, B.: Régularisation dinéquations variationelles par approximations successives. Revue Fr. Inform. Rech. Oper. 4, 154–159 (1970)

    MATH  Google Scholar 

  28. Németh, S.Z.: Monotone vector fields. Publ. Math. Debrecen 54, 437–449 (1999)

    MathSciNet  MATH  Google Scholar 

  29. Qin, X., Cho, Y.J., Kang, S.M.: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 225, 20–30 (2009)

    Article  MathSciNet  Google Scholar 

  30. Qin, X., Kang, S.M., Cho, Y.J.: Convergence theorems on generalized equilibrium problems and fixed point problems with applications. Proc. Estonian Acad. Sci. 58, 170–318 (2009)

    Article  MathSciNet  Google Scholar 

  31. Quoc, T.D., Muu, L.D., Nguyen, V.H.: Extragradient methods extended to equilibrium problems. Optimization. 57, 749–776 (2008)

    Article  MathSciNet  Google Scholar 

  32. Rahimi, R., Farajzadeh, A.P., Vaezpour, S.M.: Study of equilibrium problem in Hadamard manifolds by extending some concepts of nonlinear analysis. RACSAM 112(4), 1521–1537 (2018)

    Article  MathSciNet  Google Scholar 

  33. Ranjbar, S., Khatibzadeh, H.: \(\Delta \)-convergence and \(W\)-convergence of the modified Mann iteration for a family of asymptotically nonexpansive tipy mappings in complete CAT(0) spaces. Fixed Point Theory. 17, 151–158 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Ranjbar, S., Khatibzadeh, H.: Strong and \(\Delta \)-convergence to a zero of a monotone operator in CAT(0) Spaces. Mediterr. J. Math. 14(2), 15 (2017)

    Article  MathSciNet  Google Scholar 

  35. Reich, S., Salinas, Z.: Infinite products of discontinuous operators in Banach and metric spaces. Linear Nonlinear Anal. 1, 169–200 (2015)

    MathSciNet  MATH  Google Scholar 

  36. Reich, S., Salinas, Z.: Metric convergence of infinite products of operators in Hadamard spaces. J. Nonlinear Convex Anal. 18, 331–345 (2017)

    MathSciNet  Google Scholar 

  37. Xu, H.K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 65, 109–113 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

    R. Moharami & G. Zamani Eskandani

Authors
  1. R. Moharami
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Zamani Eskandani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Moharami.

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

Moharami, R., Eskandani, G.Z. An extragradient algorithm for solving equilibrium problem and zero point problem in Hadamard spaces. RACSAM 114, 152 (2020). https://doi.org/10.1007/s13398-020-00885-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-020-00885-5

Keywords

Mathematics Subject Classification

Search

Navigation