Skip to main content

Advertisement

SpringerLink
Log in

On nonconvex bifunction variational inequalities

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper, we introduce and consider a new class of variational inequalities, which is called the nonconvex bifunction variational inequality. We suggest and analyze some iterative methods for solving nonconvex bifunction variational inequalities using the auxiliary principle technique. We prove that the convergence of implicit method requires only pseudomonotonicity, which is weaker condition than monotonicity. Our proof of convergence is very simple. Results proved in this paper may stimulate further research in this dynamic field.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bounkhel M., Tadji L., Hamdi A.: Iterative schemes to solve nonconvex variational problems. J. Inequal. Pure Appl. Math. 4, 1–14 (2003)

    Google Scholar 

  2. Crespi G.P., Ginchev J., Rocca M.: Minty variational inequalities, increase along rays property and optimization. J. Optim. Theory Appl. 123, 479–496 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Crespi G.P., Ginchev J., Rocca M.: Existence of solutions and star-shapedness in Minty variational inequalities. J. Global Optim. 32, 485–494 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Crespi G.P., Ginchev J., Rocca M.: Increasing along rays property for vector functions. J. Nonconvex Anal. 7, 39–50 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Crespi G.P., Ginchev J., Rocca M.: Some remarks on the Minty vector variational principle. J. Math. Anal. Appl. 345, 165–175 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Clarke F.H., Ledyaev Y.S., Wolenski P.R.: Nonsmooth Analysis and Control Theory. Springer-Verlag, Berlin (1998)

    MATH  Google Scholar 

  7. Fang Y.P., Hu R.: Parametric well-posedness for variational inequalities defined by bifunction, Computer. Math. Appl. 53, 1306–1316 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Glowinski R., Lions J.L., Tremolieres R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam, Holland (1981)

    MATH  Google Scholar 

  9. Giannessi F., Maugeri A., Pardalos P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer Academics Publishers, Dordrecht (2001)

    Google Scholar 

  10. Gilbert R.P., Panagiotopoulos P.D., Pardalos P.M.: From Convexity to Nonconvexity. Kluwer Academic Publishers, Holland (2001)

    Book  MATH  Google Scholar 

  11. Kinderlehrer D., Stampacchia G.: An Introduction to Variational Inequalities and Their Applications. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  12. Lalitha C.S., Mehra M.: Vector variational inequalities with cone-pseudomonotone bifunction. Optim. 54, 327–338 (2005)

    Article  MATH  Google Scholar 

  13. Noor M.A.: General variational inequalities. Appl. Math. Letters 1, 119–121 (1988)

    Article  MATH  Google Scholar 

  14. Noor M.: New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, 217–229 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Noor M.A.: Some developments in general variational inequalities. Appl. Math. Computation 152, 199–277 (2004)

    Article  MATH  Google Scholar 

  16. Noor M.A.: Iterative schemes for nonconvex variational inequalities. J. Optim. Theory Appl. 121, 385–395 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Noor M.A.: Extended general variational inequalities Appl. Math. Letters 22, 182–185 (2009)

    Article  MATH  Google Scholar 

  18. Noor M.A.: Projection methods for nonconvex variational inequalities. Optim. Letters 3, 411–418 (2009)

    Article  MATH  Google Scholar 

  19. Noor M.A.: Implicit Iterative Methods for Nonconvex Variational Inequalities. J. Optim. Theory Appl. 143, 619–624 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Noor M.A.: An extragradient algorithm for solving general nonconvex variational inequalities. Appl. Math. Letters, 23, 917–921 (2010)

    Article  MATH  Google Scholar 

  21. Noor M.A.: On an implicit method for nonconvex variational inequalities. J. Optim. Theory Appl. 147, 411–417 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Noor M.A.: Some new classes of nonconvex functions. Nonl. Anal. Funct. Appl. 11, 165–171 (2006)

    MATH  Google Scholar 

  23. Noor M.A., Al-Said S., Noor K.I., Yao Y.: Extragradient methods for solving nonconvex variational inequalities. J. Comput. Appl. Math. 235, 3104–3108 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Noor M.A., Noor K.I.: Iterative schemes for trifunction hemivariational inequalities. Optim. Letters. 5, 273–282 (2011)

    Article  MATH  Google Scholar 

  25. Noor M.A., Noor K.I., Al-Said E.: Auxiliary principle technique for solving bifunction variational inequalities. J. Optim. Theory Appl. 149, 441–445 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Noor M.A., Noor K.I., Huang Z.Y.: Bifunction hemivariational inequalities. J. Appl. Math. Computing. 35, 595–605 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Noor M.A., Noor K.I., Rassias Th.M.: Some aspects of variational inequalities. J. Comput. Appl. Math. 47, 285–312 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear Analysis and Variational Problems. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  29. Poliquin R.A., Rockafellar R.T., Thibault L.: Local differentiability of distance functions. Trans. Amer. Math. Soc. 352, 5231–5249 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Stampacchia G.: Formes bilineaires coercitives sur les ensembles convexes. C. R. Acad. Sci. Paris. 258, 4413–4416 (1964)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan

    Muhammad Aslam Noor & Khalida Inayat Noor

  2. Mathematics Department, College of Science, King Saud University, Riyadh, Saudi Arabia

    Muhammad Aslam Noor & Eisa Al-Said

Authors
  1. Muhammad Aslam Noor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Khalida Inayat Noor
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eisa Al-Said
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Muhammad Aslam Noor.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Noor, M.A., Noor, K.I. & Al-Said, E. On nonconvex bifunction variational inequalities. Optim Lett 6, 1477–1484 (2012). https://doi.org/10.1007/s11590-011-0342-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-011-0342-y

Keywords

Search

Navigation