On gradient projection methods for strongly pseudomonotone variational inequalities without Lipschitz continuity

  • Trinh Ngoc HaiEmail author
Original Paper


In this paper, we propose two new self-adaptive algorithms for solving strongly pseudomonotone variational inequalities. Our algorithms use dynamic step-sizes, chosen based on information of the previous step and their strong convergence is proved without the Lipschitz continuity of the underlying mappings. In contrast to the existing self-adaptive algorithms in Bello Cruz et al. (Comput Optim Appl 46:247–263, 2010), Santos et al. (Comput Appl Math 30:91–107, 2011), which use fast diminishing step-sizes, the new algorithms use arbitrarily slowly diminishing step sizes. This feature helps to speed up our algorithms. Moreover, we provide the worst case complexity bound of the proposed algorithms, which have not been done in the previous works. Some preliminary numerical experiences and comparisons are also reported.


Variational inequality Complexity bound Equilibrium problem Strong pseudomonotonicity 



This research is funded by the Hanoi University of Science and Technology (HUST) under project number T2018-PC-121. The author thank two anonymous referees and the editor for their constructive comments which helped to improve the paper.


  1. 1.
    Anh, P.K., Hai, T.N.: Splitting extragradient-like algorithms for strongly pseudomonotone equilibrium problems. Numer. Algorithms 76, 67–91 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anh, P.N., Hai, T.N., Tuan, P.M.: On ergodic algorithms for equilibrium problems. J. Global Optim. 64, 179–195 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anh, P.K., Vinh, N.T.: Self-adaptive gradient projection algorithms for variational inequalities involving non-Lipschitz continuous operators. Numer. Algorithms.
  4. 4.
    Bao, T.Q., Khanh, P.Q.: A projection-type algorithm for pseudomonotone nonlipschitzian multi-valued variational inequalities. Nonconvex Optim. Appl. 77, 113–129 (2005)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bello Cruz, J.Y., Iusem, A.N.: Convergence of direct methods for paramonotone variational inequalities. Comput. Optim. Appl. 46, 247–263 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bello Cruz, J.Y., Iusem, A.N.: A strongly convergent direct method for monotone variational inequalities in Hilbert spaces. Numer. Funct. Anal. Optim. 30, 23–36 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bello Cruz, J.Y., Iusem, A.N.: An explicit algorithm for monotone variational inequalities. Optimization 61, 855–871 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Facchinei, F., Pang, J.-S.: Finite-dimensional variational inequalities and complementarity problems. Springer, New York (2003)zbMATHGoogle Scholar
  11. 11.
    Hai, T.N., Vinh, N.T.: Two new splitting algorithms for equilibrium problems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 111, 1051–1069 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Iiduka, H.: A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping. Optimization 59, 873–885 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Iiduka, H.: Fixed point optimization algorithm and its application to power control in CDMA data networks. Math. Program. 133, 227–242 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Iiduka, H., Yamada, I.: An ergodic algorithm for the power-control games for CDMA data networks. J. Math. Model. Algorithms 8, 1–18 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Khanh, P.D., Nhut, M.B.: Error bounds for strongly monotone and Lipschitz continuous variational inequalities. Optim. Lett. 12, 971–984 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Khanh, P.D., Vuong, P.T.: Modified projection method for strongly pseudomonotone variational inequalities. J. Global Optim. 58, 341–350 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kim, D.S., Vuong, P.T., Khanh, P.D.: Qualitative properties of strongly pseudomonotone variational inequalities. Optim. Lett. 10, 1669–1679 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  19. 19.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Mat. Metody. 12, 747–756 (1976)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Malitsky, Y.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502–520 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Santos, P., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30, 91–107 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Solodov, M.V.: Merit functions and error bounds for generalized variational inequalities. J. Math. Anal. Appl. 287, 405–414 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Solodov, M.V., Svaiter, B.F.: A new projection method for monotone variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Thuy, L.Q., Hai, T.N.: A projected subgradient algorithm for bilevel equilibrium problems and applications. J. Optim. Theory Appl. 175, 411–431 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Applied Mathematics and InformaticsHanoi University of Science and TechnologyHanoiVietnam

Personalised recommendations