Advertisement

Computational Optimization and Applications

, Volume 66, Issue 1, pp 75–96 | Cite as

Modified hybrid projection methods for finding common solutions to variational inequality problems

  • Dang Van Hieu
  • Pham Ky Anh
  • Le Dung MuuEmail author
Article

Abstract

In this paper we propose several modified hybrid projection methods for solving common solutions to variational inequality problems involving monotone and Lipschitz continuous operators. Based on differently constructed half-spaces, the proposed methods reduce the number of projections onto feasible sets as well as the number of values of operators needed to be computed. Strong convergence theorems are established under standard assumptions imposed on the operators. An extension of the proposed algorithm to a system of generalized equilibrium problems is considered and numerical experiments are also presented.

Keywords

Variational inequality Equilibrium problem Generalized equilibrium problem Gradient method Extragradient method 

Mathematics Subject Classification

65Y05 65K15 68W10 47H05 47H10 

Notes

Acknowledgments

The authors would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions which helped us very much in improving the original version of this paper. The work of the second and third authors is supported by VIASM.

References

  1. 1.
    Alber, Y., Ryazantseva, I.: Nonlinear Ill-Posed Problems of Monotone Type. Spinger, Dordrecht (2006)zbMATHGoogle Scholar
  2. 2.
    Alizadeh, M.H., Bianchi, M., Hadjisavvas, N., Pini, R.: On cyclic and \(n\)-cyclic monotonicity of bifunctions. J. Glob. Optim. 60, 599–616 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anh, P.K., Buong, N., Hieu, D.V.: Parallel methods for regularizing systems of equations involving accretive operators. Appl. Anal. 93, 2136–2157 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Anh, P.K., Hieu, D.V.: Parallel and sequential hybrid methods for a finite family of asymptotically quasi \(\phi \)-nonexpansive mappings. J. Appl. Math. Comput. 48, 241–263 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Anh, P.K., Hieu, D.V.: Parallel hybrid methods for variational inequalities, equilibrium problems and common fixed point problems. Vietnam J. Math. 44(2), 351–374 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bartz, S., Bauschke, H.H., Borwein, J.M., Reich, S., Wang, X.: Fitzpatrick functions, cyclic monotonicity and Rockafellar’s antiderivative. Nonlinear Anal. 66, 1198–1223 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Censor, Y., Gibali, A., Reich, S., Sabach, S.: Common solutions to variational inequalities. Set-Valued Var. Anal. 20, 229–247 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Censor, Y., Gibali, A., Reich, S.: A von Neumann alternating method for finding common solutions to variational inequalities. Nonlinear Anal. 75, 4596–4603 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Methods Softw. 26(4–5), 827–845 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevichs extragradient method for the variational inequality problem in Euclidean space. Optimization 61, 1119–1132 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, Berlin (2003)zbMATHGoogle Scholar
  18. 18.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York and Basel (1984)zbMATHGoogle Scholar
  19. 19.
    Harker, P.T., Pang, J.-S.: A damped-newton method for the linear complementarity problem. Lect. Appl. Math. 26, 265–284 (1990)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hartman, P., Stampacchia, G.: On some non-linear elliptic diferential-functional equations. Acta Math. 115, 271–310 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hieu, D.V.: A parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive mappings in Hilbert space. J. Korean Math. Soc. 52, 373–388 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms (2016). doi: 10.1007/s11075-015-0092-5
  23. 23.
    Hieu, D.V.: Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings. J. Appl. Math. Comput. (2016). doi: 10.1007/s12190-015-0980-9
  24. 24.
    Hieu, D.V.: Parallel extragradient-proximal methods for split equilibrium problems. Math. Model. Anal. (2016). doi: 10.3846/13926292.2016.1183527
  25. 25.
    Kassay, G., Reich, S., Sabach, S.: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 21, 1319–1344 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  27. 27.
    Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000)zbMATHGoogle Scholar
  28. 28.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Mat. Metody. 12, 747–756 (1976)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, New York (1971)CrossRefzbMATHGoogle Scholar
  30. 30.
    Malitsky, Y.V.: Projected reflected gradient methods for monotone variational inequalities. SIAM J. Optim. 25, 502–520 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Glob. Optim. 61, 193–202 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Peng, J.W., Yao, J.C.: Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudocontractions and monotone mappings. Taiwan. J. Math. 13, 1537–1582 (2009)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Peng, J.W., Yao, J.C.: Two extragradient methods for generalized mixed equilibrium problems, nonexpansive mappings and monotone mappings. Comput. Math. Appl. 58, 1287–1301 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Petrot, N., Wattanawitoon, K., Kumam, P.: A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces. Nonlinear Anal.: Hybrid Syst. 4, 631–643 (2010)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Takahashi, S., Takahashi, W.: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69, 1025–1033 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Vilenkin, N.Y., Gorin, E.A., Kostyuchenko, A.G., Krasnosel’skii, M.A., Krein, S.G., Maslov, V.P., Mityagin, B.S., Petunin, Y., et al.: Functional Analysis. Wolters-Noordhoff, Groningen (1972)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsVietnam National University, HanoiHanoiVietnam
  2. 2.Institute of MathematicsVAST, HanoiHanoiVietnam

Personalised recommendations