Advertisement

Linesearch methods for bilevel split pseudomonotone variational inequality problems

  • Tran Viet Anh
Original Paper
  • 15 Downloads

Abstract

In this paper, we propose Linesearch methods for solving a bilevel split variational inequality problem (BSVIP) involving a strongly monotone mapping in the upper-level problem and pseudomonotone mappings in the lower-level one. A strongly convergent algorithm for such a BSVIP is proposed and analyzed.

Keywords

Bilevel split variational inequality problem Linesearch methods Pseudomonotone mapping Strong convergence 

Mathematics Subject Classification (2010)

47J25 47N10 90C25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author is very grateful to the reviewers for useful comments and advices which helped to improve the quality of this paper.

Funding information

The research of the author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2017.315.

References

  1. 1.
    Anh, P.K., Anh, T.V., Muu, L.D.: On bilevel split pseudomonotone variational inequality problems with applications. Acta Math. Vietnam. 42, 413–429 (2017)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anh, P.K., Hieu, D.V.: Parallel hybrid methods for variational inequalities, equilibrium problems and common fixed point problems. Vietnam J. Math. 44, 351–374 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Anh, P.N., Hien, N.D.: Hybrid proximal point and extragradient algorithms for solving equilibrium problems. Acta Math. Vietnam. 39, 405–423 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Anh, T.V.: A strongly convergent subgradient Extragradient-Halpern method for solving a class of bilevel pseudomonotone variational inequalities. Vietnam J. Math. 45, 317–332 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Anh, T.V.: An extragradient method for finding minimum-norm solution of the split equilibrium problem. Acta Math. Vietnam. 42, 587–604 (2017)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Anh, T.V.: A parallel method for variational inequalities with the multiple-sets split feasibility problem constraints. J. Fixed Point Theory Appl. 19, 2681–2696 (2017)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Anh, T.V., Muu, L.D.: A projection-fixed point method for a class of bilevel variational inequalities with split fixed point constraints. Optimization 65, 1229–1243 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Buong, N.: Iterative algorithms for the multiple-sets split feasibility problem in Hilbert spaces. Numer. Algorithms 76, 783–798 (2017)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ceng, L.C., Ansari, Q.H., Yao, J.C.: Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem. Nonlinear Anal. 75, 2116–2125 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)CrossRefGoogle Scholar
  12. 12.
    Censor, Y., Segal, A.: Iterative projection methods in biomedical inverse problems. In: Censor, Y, Jiang, M., Louis, A.K. (eds.) Mathematical Methods in Biomedical Imaging and Intensity-Modulated Therapy, IMRT, Edizioni della Norale, Pisa, pp. 65–96 (2008)Google Scholar
  13. 13.
    Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Prob. 21, 2071–2084 (2005)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)MathSciNetMATHGoogle Scholar
  16. 16.
    Dinh, B.V., Muu, L.D.: Algorithms for a class of bilevel programs involving pseudomonotone variational inequalities. Acta Math. Vietnam. 38, 529–540 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Facchinei, F., Pang, J.S.: Finite-dimensional variational inequalities and complementary problems. Springer, New York (2003)MATHGoogle Scholar
  18. 18.
    Goebel, K., Kirk, W.A.: Topics in metric fixed point theory. In: Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press (1990)Google Scholar
  19. 19.
    Khanh, P.D.: Convergence rate of a modified extragradient method for pseudomonotone variational inequalities. Vietnam J. Math. 45, 397–408 (2017)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Konnov, I.V.: Combined relaxation methods for variational inequalities. Springer, Berlin (2000)MATHGoogle Scholar
  21. 21.
    Liu, Y.: A modified hybrid method for solving variational inequality problems in Banach spaces. J. Nonlinear Funct. Anal. 2017, Article ID 31 (2017)Google 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)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Oettli, W.: A remark on vector valued equilibria and generalized monotonicity. Acta Math. Vietnam. 22, 213–221 (1997)MathSciNetMATHGoogle Scholar
  24. 24.
    Qin, X., Yao, J.C.: Projection splitting algorithm for nonself operator. J. Nonlinear Convex Anal. 18, 925–935 (2017)MathSciNetMATHGoogle Scholar
  25. 25.
    Tran, D.Q., Muu, L.D., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Sahu, D.R., Yao, J.C.: A generalized hybrid steepest descent method and applications. J. Nonlinear Var. Anal. 1, 111–126 (2017)Google Scholar
  27. 27.
    Xu, H.K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66, 240–256 (2002)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Yamada, I.: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms for Feasibility and Optimization and their Applications, pp. 473–504. Elsevier, New York (2001)Google Scholar
  29. 29.
    Yu, X., Shahzad, N., Yao, Y.: Implicit and explicit algorithms for solving the split feasibility problem. Optim. Lett. 6, 1447–1462 (2012)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Scientific FundamentalsPosts and Telecommunications Institute of TechnologyHanoiVietnam

Personalised recommendations