Advertisement

Journal of Optimization Theory and Applications

, Volume 138, Issue 2, pp 253–273 | Cite as

Solution Methods for Pseudomonotone Variational Inequalities

  • N. N. Tam
  • J. C. YaoEmail author
  • N. D. Yen
Article

Abstract

We extend some results due to Thanh-Hao (Acta Math. Vietnam. 31: 283–289, [2006]) and Noor (J. Optim. Theory Appl. 115:447–452, [2002]). The first paper established a convergence theorem for the Tikhonov regularization method (TRM) applied to finite-dimensional pseudomonotone variational inequalities (VIs), answering in the affirmative an open question stated by Facchinei and Pang (Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, [2003]). The second paper discussed the application of the proximal point algorithm (PPA) to pseudomonotone VIs. In this paper, new facts on the convergence of TRM and PPA (both the exact and inexact versions of PPA) for pseudomonotone VIs in Hilbert spaces are obtained and a partial answer to a question stated in (Acta Math. Vietnam. 31:283–289, [2006]) is given. As a byproduct, we show that the convergence theorem for inexact PPA applied to infinite-dimensional monotone variational inequalities can be proved without using the theory of maximal monotone operators.

Keywords

Variational inequalities Pseudomonotone operators Tikhonov regularization method Proximal point algorithms Convergence 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Thanh-Hao, N.: Tikhonov regularization algorithm for pseudomonotone variational inequalities. Acta Math. Vietnam. 31, 283–289 (2006) MathSciNetGoogle Scholar
  2. 2.
    Noor, M.A.: Proximal methods for mixed variational inequalities. J. Optim. Theory Appl. 115, 447–452 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I, II. Springer, New York (2003) Google Scholar
  4. 4.
    Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.): Handbook of Generalized Convexity and Generalized Monotonicity. Springer, New York (2005) zbMATHGoogle Scholar
  5. 5.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 127–149 (1994) MathSciNetGoogle Scholar
  6. 6.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, San Diego (1980) zbMATHGoogle Scholar
  7. 7.
    Aussel, D., Hadjisavvas, N.: On quasimonotone variational inequalities. J. Optim. Theory Appl. 121, 445–450 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Crouzeix, J.-P.: Pseudomonotone varational inequality problems: existence of solutions. Math. Program. 78, 305–314 (1997) MathSciNetGoogle Scholar
  9. 9.
    El Farouq, N.: Pseudomonotone variational inequalities: convergence of the auxiliary problem method. J. Optim. Theory Appl. 111, 305–326 (2001). Errata Corrige: 114, 477 (2002) CrossRefMathSciNetGoogle Scholar
  10. 10.
    El Farouq, N.: Convergent algorithm based on progressive regularization for solving pseudomonotone variational inequalities. J. Optim. Theory Appl. 120, 455–485 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    El Farouq, N., Cohen, G.: Progressive regularization of variational inequalities and decomposition algorithms. J. Optim. Theory Appl. 97, 407–433 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Konnov, I.V.: Application of the proximal point method to nonmonotone equilibrium problems. J. Optim. Theory Appl. 119, 317–333 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Konnov, I.V., Ali, M.S.S., Mazurkevich, E.O.: Regulization for nonmonotone variational inequalities. Appl. Math. Optim. 53, 311–330 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Noor, M.A.: Proximal methods for mixed quasivariational inequalities. J. Optim. Theory Appl. 115, 453–459 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Noor, M.A.: Modified projection method for pseudomonotone variational inequalities. Appl. Math. Lett. 15, 315–320 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Noor, M.A.: Pseudomonotone general mixed variational inequalities. Appl. Math. Comput. 141, 529–540 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Noor, M.A.: Resolvent algorithms for mixed quasivariational inequalities. J. Optim. Theory Appl. 119, 137–149 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Noor, M.A.: Projection-proximal methods for general variational inequalities. J. Math. Anal. Appl. 318, 53–62 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Yao, J.-C.: Multi-valued variational inequalities with K-pseudomonotone operators. J. Optim. Theory Appl. 80, 63–74 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kien, B.T., Yao, J.-C., Yen, N.D.: On the solution existence of pseudomonotone variational inequalities. Institute of Mathematics, Hanoi, E-Preprint 2006/10/01. J. Glob. Optim. 41, 135–145, 2008 Google Scholar
  21. 21.
    Noor, M.A.: Auxiliary principle technique for equilibrium problems. J. Optim. Theory Appl. 122, 371–386 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Zeng, L.-C., Lin, L.J., Yao, J.-C.: Auxiliary problem method for mixed variational-like inequalities. Taiwan. J. Math. 10, 497–513 (2006) zbMATHMathSciNetGoogle Scholar
  23. 23.
    Schaible, S., Yao, J.-C., Zeng, L.-C.: A proximal method for pseudomonotone type variational-like inequalities. Taiwan. J. Math. 10, 497–513 (2006) zbMATHMathSciNetGoogle Scholar
  24. 24.
    Martinet, B.: Régularisation d’inéquations variationnelles par approximations successives. Rev. Fr. Inform. Rech. Opér. 4, 154–158 (1970) MathSciNetGoogle Scholar
  25. 25.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Yen, N.D., Lee, G.M.: Some remarks on the elliptic regularization method. In: Cho, Y.J. (ed.) Fixed Point Theory and Applications, pp. 127–134. Nova Science Publishers, New York (2000) Google Scholar
  27. 27.
    Yao, J.-C., Chadli, O.: Pseudomonotone complementarity problems and variational inequalities. In: Hadjisavvas, N., Komlósi, S., Schaible, S. (eds.) Handbook of Generalized Convexity and Generalized Monotonicity, pp. 501–558. Springer, New York (2005) CrossRefGoogle Scholar
  28. 28.
    Brezis, H.: Analyse Fonctionnelle, 2nd edn. Masson, Paris (1987) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsHanoi University of Pedagogy No. 2Phuc YenVietnam
  2. 2.Department of Applied MathematicsNational Sun Yat-Sen UniversityKaohsiungTaiwan
  3. 3.Institute of MathematicsVietnamese Academy of Science and TechnologyHanoiVietnam

Personalised recommendations