Skip to main content
Log in

Convergence analysis of the extragradient method for equilibrium problems in Hadamard spaces

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

We propose an extragradient method for solving equilibrium problems of pseudo-monotone type in Hadamard spaces. We prove \(\Delta \)-convergence of the generated sequence to a solution of the equilibrium problem, under standard assumptions on the bifunction. Then, we propose a regularization procedure which ensures strong convergence of the generated sequence to an equilibrium point of the problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Ahmadi Kakavandi B (2013) Weak topologies in complete CAT(0) spaces. Proc Am Math Soc 141:1029–1039

    Article  MathSciNet  Google Scholar 

  • Ahmadi Kakavandi B, Amini M (2010) Duality and subdifferential for convex function on CAT(0) metric spaces. Nonlinear Anal 73:3450–3455

    Article  MathSciNet  Google Scholar 

  • Armijo L (1966) Minimization of functions having continuous partial derivatives. Pac J Math 16:1–3

    Article  MathSciNet  Google Scholar 

  • Bacak M (2014) Convex analysis and optimization in Hadamard spaces. De Gruyter, Berlin

    Book  Google Scholar 

  • Bao TQ, Khanh PQ (2005) A projection-type algorithm for pseudomonotone nonlipschitzian multivalued variational inequalities. Nonconvex Optim Appl 77:113–129

    Article  MathSciNet  Google Scholar 

  • Batista EE, Bento GC, Ferreira OP (2019) An extragradient algorithm for variational inequality in Hadamard manifolds. To be published in ESAIM, Control, Optimisation and Calculus of Variations

  • Berg ID, Nikolaev IG (1998) On a distance between directions in an Alexandrov space of curvature \(\le \) K. Mich Math J 45:275–289

    Article  Google Scholar 

  • Berg ID, Nikolaev IG (2008) Quasilinearization and curvature of Alexandrov spaces. Geom Dedic 133:195–218

    Article  Google Scholar 

  • Bertrand J, Kloeckner B (2012) A geometric study of Wasserstein spaces: Hadamard spaces. J Topol Anal 4:515–542

    Article  MathSciNet  Google Scholar 

  • Bertsekas DP, Tsitsiklis JN (1989) Parallel and distributed computation: numerical methods. Prentice Hall, New Jersey

    MATH  Google Scholar 

  • Bianchi M, Schaible S (1996) Generalized monotone bifunctions and equilibrium problems. J Optim Theory Appl 90:31–43

    Article  MathSciNet  Google Scholar 

  • Chadli O, Chbani Z, Riahi H (2000) Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities. J Optim Theory Appl 105:299–323

    Article  MathSciNet  Google Scholar 

  • Colao V, López G, Marino G, Martín-Márquez V (2012) Equilibrium problems in Hadamard manifolds. J Math Anal Appl 388:61–77

    Article  MathSciNet  Google Scholar 

  • Combettes PL, Hirstoaga SA (2005) Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal 6:117–136

    MathSciNet  MATH  Google Scholar 

  • Dehghan H, Rooin J (2013) A characterization of metric projection in Hadamard spaces with applications. J Nonlinear Convex Anal (to be published)

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

    Article  MathSciNet  Google Scholar 

  • Fang SC (1980) An iterative method for generalized complementarity problems. IEEE Trans Autom Control 25:1225–1227

    Article  MathSciNet  Google Scholar 

  • Ferreira OP, Lucambio Pérez LR, Németh SZ (2005) Singularities of monotone vector fields and extragradient algorithm. J Global Optim 31:133–151

    Article  MathSciNet  Google Scholar 

  • Gárciga Otero R, Iusem AN, Svaiter BF (2001) On the need for hybrid steps in hybrid proximal point methods. Oper Res Lett 29:217–220

    Article  MathSciNet  Google Scholar 

  • Halpern B (1967) Fixed points of nonexpansive mappings. Bull Am Math Soc 73:957–961

    Article  Google Scholar 

  • Iusem AN, Lucambio Pérez LR (2000) An extragradient-type algorithm for non-smooth variational inequalities. Optimization 48:309–332

    Article  MathSciNet  Google Scholar 

  • Iusem AN, Mohebbi V (2018) Extragradient method for nonsmooth equilibrium problems in Banach spaces. Optimization. https://doi.org/10.1080/02331934.2018.1462808

    Article  MATH  Google Scholar 

  • Iusem AN, Nasri M (2011) Korpolevich’s method for variational inequality problems in Banach spaces. J Global Optim 50:59–76

    Article  MathSciNet  Google Scholar 

  • Iusem AN, Sosa W (2010) On the proximal point method for equilibrium problems in Hilbert spaces. Optimization 59:1259–1274

    Article  MathSciNet  Google Scholar 

  • Iusem AN, Svaiter BF (1997) A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42:309–321

    Article  MathSciNet  Google Scholar 

  • Iusem AN, Kassay G, Sosa W (2009) On certain conditions for the existence of solutions of equilibrium problems. Math Program 116:259–273

    Article  MathSciNet  Google Scholar 

  • Jost J (1997) Nonpositive Curvature: geometric and analytic aspects. Lectures in mathematics ETH Zürich. Birkhäuser, Basel

    Book  Google Scholar 

  • Khatibzadeh H, Mohebbi V (2016) Proximal point algorithm for infinite pseudo-monotone bifunctions. Optimization 65:1629–1639

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Khatibzadeh H, Mohebbi V, Ranjbar S (2017) New results on the proximal point algorithm in non-positive curvature metric spaces. Optimization 66:1191–1199

    Article  MathSciNet  Google Scholar 

  • Khobotov EN (1987) Modifications of the extragradient method for solving variational inequalities and certain optimization problems. USSR Comput Math Math Phys 27:120–127

    Article  MathSciNet  Google Scholar 

  • Kirk WA (2007) Some recent results in metric fixed point theory. J Fixed Point Theory Appl 2:195–207

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Konnov IV (1993) Combined relaxation methods for finding equilibrium points and solving related problems. Rus Math 37:34–51

    MATH  Google Scholar 

  • Konnov IV (1993) On combined relaxation methods’ convergence rates. Rus Math 37:89–92

    MATH  Google Scholar 

  • Konnov IV (2001) Combined relaxation methods for variational inequalities. Springer, Berlin

    Book  Google Scholar 

  • Korpelevich GM (1976) The extragradient method for finding saddle points and other problems. Ekonomika i Matematcheskie Metody 12:747–756

    MathSciNet  MATH  Google Scholar 

  • Marcotte P (1991) Application of Khobotov’s algorithm to variational inequalities and network equilibrium problems. Inf Syst Oper Res 29:258–270

    MATH  Google Scholar 

  • Noor MA, Noor KI (2012) Some algorithms for equilibrium problems on Hadamard manifolds. J Inequal Appl 2012:230

    Article  MathSciNet  Google Scholar 

  • Robinson SM, Lu S (2008) Solution continuity in variational conditions. J Glob Optim 40:405–415

    Article  MathSciNet  Google Scholar 

  • Saejung S, Yotkaew P (2012) Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal 75:742–750

    Article  MathSciNet  Google Scholar 

  • Tang G-J, Huang N-J (2012) Korpelevich’s method for variational inequality problems on Hadamard manifolds. J Glob Optim 54:493–509

    Article  MathSciNet  Google Scholar 

  • Tang G-J, Wang X, Liu H-W (2015) A projection-type method for variational inequalities on Hadamard manifolds and verification of solution existence. Optimization 64:1081–1096

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Research for this paper by the second author was supported by CNPq and IMPA. The second author is grateful to CNPq and IMPA for his post-doctoral scholarship.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alfredo N. Iusem.

Additional information

Communicated by Ernesto G. Birgin.

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

Iusem, A.N., Mohebbi, V. Convergence analysis of the extragradient method for equilibrium problems in Hadamard spaces. Comp. Appl. Math. 39, 44 (2020). https://doi.org/10.1007/s40314-020-1076-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-020-1076-1

Keywords

Mathematics Subject Classification

Navigation