Acta Mathematica Vietnamica

, Volume 42, Issue 1, pp 61–79 | Cite as

Outer-Inner Approximation Projection Methods for Multivalued Variational Inequalities

  • Pham Ngoc AnhEmail author
  • Le Thi Hoai An


In this paper, we present new projection methods for solving multivalued variational inequalities on a given nonlinear convex feasible domain. The first one is an extension of the extragradient method to multivalued variational inequalities under the asymptotic optimality condition, but it must satisfy certain Lipschitz continuity conditions. To avoid this requirement, we propose linesearch procedures commonly used in variational inequalities to obtain an approximation linesearch method for solving multivalued variational inequalities. Next, basing on a family of nonempty closed convex subsets of \(\mathcal R^{n}\) and linesearch techniques, we give inner approximation projection algorithms for solving multivalued variational inequalities and the convergence of the algorithms is established under few assumptions.


Multivalued variational inequalities Upper semicontinuous Linesearch Projection method 

Mathematics Subject Classification (2010)

65K10 90C25 



We are very grateful to the anonymous referees for their really helpful and constructive comments. The work presented here was completed while the first author was on leave at LITA, University of Lorraine, France. He wishes to thank the Fonds Europeens de Developpement Regional Lorraine for the financial support via the FEDER project ”INNOMAD”. This work was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED).


  1. 1.
    Anh, P.N.: An interior proximal method for solving pseudomonotone nonlipschitzian multivalued variational inequalities. Nonl. Anal. Forum 14, 27–42 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anh, P.N., Kim, J.K.: A new method for solving monotone generalized variational inequalities. J. Ineq. Appl. 2010 (2010). doi: 10.1155/2010/657192. Article ID 657192
  3. 3.
    Anh, P.N., Kim, J.K.: The interior proximal cutting hyperplane method for multivalued variational inequalities. J. Nonl. Conv. Anal. 11, 491–502 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Anh, P.N., Kuno, T.: A cutting hyperplane method for generalized monotone nonlipschitzian multivalued variational inequalities. In: Bock, H. G., Phu, H. X., Rannacher, R., Schloder, J. P. (eds.) Modeling, Simulation and Optim.ization of Complex Processes. Springer, Berlin (2012)Google Scholar
  5. 5.
    Anh, P.N., Muu, L.D.: Contraction mapping fixed point algorithms for multivalued mixed variational inequalities on network. In: Dempe, S., Vyacheslav, K. (eds.) Optim.ization with Multivalued Mappings. Springer (2006)Google Scholar
  6. 6.
    Anh, P.N., Muu, L.D., Strodiot, J.J.: Generalized projection method for non-Lipschitz multivalued monotone variational inequalities. Acta Math. Vietnam. 34, 67–79 (2009)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Anh, P.N., Muu, L.D., Nguyen, V.H., Strodiot, J.J.: On the contraction and nonexpensiveness properties of the marginal mappings in generalized variational inequalities involving cocoercive operators. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds.) Generalized convexity and monotonicity. Springer (2005)Google Scholar
  8. 8.
    Anh, P.N., Muu, L.D., Nguyen, V.H., Strodiot, J.J.: Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities. J. Opt. Theory Appl. 124, 285–306 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Attouch, H.: Variational Convergence for Functions and Operators. Pitman, London (1984)zbMATHGoogle Scholar
  10. 10.
    Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)zbMATHGoogle Scholar
  11. 11.
    Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optim. (2010). doi: 10.1080/02331934.2010.539689 zbMATHGoogle Scholar
  12. 12.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementary Problems. Springer, New York (2003)zbMATHGoogle Scholar
  13. 13.
    Fanga, C., He, Y.: A double projection algorithm for multivalued variational inequalities and a unified framework of the method. Appl. Math. Comp. 217, 9543–9551 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Iiduka, H., Yamada, I.: A subgradient-type method for the equilibrium problem over the fixed point set and its applications. Optim. 58, 251–261 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Iiduka, H., Yamada, I.: An ergodic algorithm for the power-control games for CDMA data networks. J. Math. Model. Algor. 8, 1–18 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optim. 52, 301–316 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)zbMATHGoogle Scholar
  19. 19.
    Taji, K., Fukushima, M.: A new merit function and a successive quadratic programming algorithm for variational inequality problems. SIAM J. Optim. 6, 704–713 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl 331, 506–515 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Strodiot, J.J., Nguyen, T.T.V., Nguyen, V.H.: A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems. J. Global Optim. (2012). doi: 10.1007/s10898-011-9814-y zbMATHGoogle Scholar
  23. 23.
    Yao, Y., Noor, M.A., Liou, Y.C., Kang, S.M.: Iterative algorithms for general multivalued variational inequalities. Abstract Appl. Anal. 2012 (2012). doi: 10.1155/2012/768272. Article ID 768272
  24. 24.
    Zheng, L.: The subgradient double projection method for variational inequalities in a Hilbert space. Fixed Point Theory Appl. 2013 (2013). doi: 10.1186/1687-1812-2013-136. Article ID 2013:136

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.Laboratory of Applied Mathematics and ComputingPTITHanoiVietnam
  2. 2.Laboratory of Theoretical and Applied Computer Science-LITA EA 3097University of LorraineIle du SaulcyFrance

Personalised recommendations