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Numerical Algorithms

, Volume 82, Issue 3, pp 761–789 | Cite as

Some extragradient-viscosity algorithms for solving variational inequality problems and fixed point problems

  • Duong Viet ThongEmail author
  • Dang Van Hieu
Original Paper

Abstract

The purpose of this paper is to study and analyze two different kinds of extragradient-viscosity-type iterative methods for finding a common element of the set of solutions of the variational inequality problem for a monotone and Lipschitz continuous operator and the set of fixed points of a demicontractive mapping in real Hilbert spaces. Although the problem can be translated to a common fixed point problem, the algorithm’s structure is not derived from algorithms in this field but from the field of variational inequalities and hence can be computed quite easily. We extend several results in the literature from weak to strong convergence in real Hilbert spaces and moreover, the prior knowledge of the Lipschitz constant of cost operator is not needed. We prove two strong convergence theorems for the sequences generated by these new methods. Primary numerical examples illustrate the validity and potential applicability of the proposed schemes.

Keywords

Extragradient method Subgradient extragradient method Tseng’s method Viscosity method Variational inequality problem Fixed point problem Demicontractive mapping 

Mathematics Subject Classification (2010)

47H09 47H10 47J20 47J25 

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Notes

Acknowledgments

The authors would like to thank Dr. Aviv Gibali and two anonymous reviewers for their comments on the manuscript which helped us very much in improving and presenting the original version of this paper.

References

  1. 1.
    Antipin, A.S.: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Mat. Metody 12, 1164–1173 (1976)Google Scholar
  2. 2.
    Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms. 56, 301–323 (2012)MathSciNetzbMATHGoogle Scholar
  3. 3.
    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)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Meth. Softw. 26, 827–845 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61, 1119–1132 (2011)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ceng, L.C., Hadjisavvas, N., Wong, N.C.: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Glob. Optim. 46, 635–646 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chidume, C.E., Maruster, S.: Iterative methods for the computation of fixed points of demicontractive mappings. J. Comput. Appl. Math. 234, 861–882 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Denisov, S.V., Semenov, V.V., Chabak, L.M.: Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern. Syst. Anal. 51, 757–765 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Marcel Dekker, New York (1984)zbMATHGoogle Scholar
  10. 10.
    Harker, P.T., Pang, J.S.: A damped-Newton method for the linear complementarity problem. Lect. Appl. Math. 26, 265–284 (1990)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hicks, T.L., Kubicek, J.D.: On the Mann iteration process in a Hilbert space. J. Math. Anal. Appl. 59, 498–504 (1997)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hieu, D.V., Anh, P.K., Muu, L.D.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. 66, 75–96 (2017)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hieu, D.V., Thong, D.V.: New extragradient-like algorithms for strongly pseudomonotone variational inequalities. J. Glob. Optim. 70, 385–399 (2018)MathSciNetzbMATHGoogle Scholar
  14. 14.
    He, B.S., Liao, L.Z.: Improvements of some projection methods for monotone nonlinear variational inequalities. J. Optim. Theory Appl. 112, 111–128 (2002)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Iiduka, H., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings. Nonlinear Anal. TMA. 61, 341–350 (2005)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Khanh, P.D., Vuong, P.T.: Modified projection method for strongly pseudomonotone variational inequalities. J. Global Optim. 58, 341–350 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Khanh, P.D.: A modifed extragradient method for infnite-dimensional variational inequalities. Acta Math Vietnam. 41, 251–263 (2016)MathSciNetGoogle Scholar
  18. 18.
    Kraikaew, R., Saejung, S.: Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 163, 399–412 (2014)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody. 12, 747–756 (1976)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Liu, L.S.: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach space. J. Math. Anal. Appl. 194, 114–125 (1995)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lyashko, S.I., Semenov, V.V., Voitova, T.A.: Low-cost modification of Korpelevich’s methods for monotone equilibrium problems. Cybern. Syst. Anal. 47, 631–640 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Maingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Glob. Optim. 61, 193–202 (2015)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Malitsky, Y.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502–520 (2015)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Mann, W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Mongkolkeha, C., Cho, Y.J., Kumam, P.: Convergence theorems for k-dimeicontactive mappings in Hilbert spaces. Math. Inequal. Appl. 16, 1065–1082 (2013)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Moudafi, A.: Viscosity approximating methods for fixed point problems. J. Math. Anal. Appl. 241, 46–55 (2000)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Nadezhkina, N., Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191–201 (2006)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Reich, S.: Constructive Techniques for Accretive and Monotone Operators. Applied Nonlinear Analysis, pp 335–345. Academic Press, New York (1979)Google Scholar
  31. 31.
    Shehu, Y., Iyiola, O.S.: Strong convergence result for monotone variational inequalities. Numer. Algorithms. 76, 259–282 (2017)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417–428 (2003)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Thong, D.V.: Viscosity approximation methods for solving fixed point problems and split common fixed point problems. J. Fixed Point Theory Appl. 19, 1481–1499 (2017)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Thong, D.V., Hieu, D.V.: Weak and strong convergence theorems for variational inequality problems. Numer. Algorithms. 78, 1045–1060 (2108)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Thong, D.V., Hieu, D.V.: Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems. Optimization 67, 83–102 (2018)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Thong, D.V., Hieu, D.V.: Modified subgradient extragradient method for variational inequality problems. Numer. Algorithms.  https://doi.org/10.1007/s11075-017-0452-4 (2017)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Thong, D.V., Hieu, D.V.: Inertial extragradient algorithms for strongly pseudomonotone variational inequalities. J. Comput. Appl. Math. 341, 80–98 (2018)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Wang, F.H., Xu, H.K.: Weak and strong convergence theorems for variational inequality and fixed point problems with Tseng’s extragradient method. Taiwanese J. Math. 16, 1125–1136 (2012)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Yang, J., Liu, H.: Strong convergence result for solving monotone variational inequalities in Hilbert space, Numer Algorithms.  https://doi.org/10.1007/s11075-018-0504-4 (2018)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Yao, Y., Marino, G., Muglia, L.: A modified Korpelevich’s method convergent to the minimum-norm solution of a variational inequality. Optimization 63, 559–569 (2014)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Zeng, L.C., Yao, J.C.: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan J. Math. 10, 1293–1303 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Applied Analysis Research Group, Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Department of MathematicsCollege of Air ForceNha TrangVietnam

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