Skip to main content
Log in

A solution method for linear variational relation problems

Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, we consider a particular class of variational relation problem namely linear variational relation problem wherein the sets are defined by linear inequalities. The purpose is to study the existence of the solution set and its nature for this class of problem. Using these results, we provide algorithms to obtain the solutions of the problem based on which we present some numerical illustrations.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Agarwal, R.P., Balaj, M., O’Regan, D.: Variational relation problems in locally convex spaces. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 18, 501–512 (2011)

    Google Scholar 

  2. Agarwal, R.P., Balaj, M., O’Regan, D.: A unifying approach to variational relation problems. J. Optim. Theory Appl. 155, 417–429 (2012)

    Google Scholar 

  3. Alimohammady, M., Balooee, J., Cho, Y.J., Roohi, M.: New perturbed finite step iterative algorithms for a system of extended generalized nonlinear mixed quasi-variational inclusions. Comput. Math. Appl. 60, 2953–2970 (2010)

    Article  Google Scholar 

  4. Balaj, M.: A fixed point-equilibrium theorem with applications. Bull. Belg. Math. Soc. Simon Stevin 17, 919–928 (2010)

    Google Scholar 

  5. Balaj, M., Lin, L.J.: Equivalent forms of a generalized KKM theorem and their applications. Nonlinear Anal. 73, 673–682 (2010)

    Article  Google Scholar 

  6. Balaj, M., Lin, L.J.: Generalized variational relation problems with applications. J. Optim. Theory Appl. 148, 1–13 (2011)

    Article  Google Scholar 

  7. Balaj, M., Luc, D.T.: On mixed variational relation problems. Comput. Math. Appl. 60, 2712–2722 (2010)

    Article  Google Scholar 

  8. Barber, C.B., Dobkin, D.P., Huhdanpaa, H.T.: The Quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22, 469–483 (1996)

    Article  Google Scholar 

  9. Ding, X.P.: Algorithm of solutions for mixed implicit quasi-variational inequalities with fuzzy mappings. Comput. Math. Appl. 38, 231–241 (1999)

    Article  Google Scholar 

  10. Ding, X.P., Feng, H.R.: Algorithm for solving a new class of generalized nonlinear implicit quasi-variational inclusions in Banach spaces. Appl. Math. Comput. 208, 547–555 (2009)

    Article  Google Scholar 

  11. Farajzadeh, A.P., Amini-Harandi, A., Kazmi, K.R.: Existence of solutions to generalized vector variational-like inequalities. J. Optim. Theory Appl. 146, 95–104 (2010)

    Article  Google Scholar 

  12. Guerra Vázqueza, F., Ruckmann, J.-J., Stein, O., Still, G.: Generalized semi-infinite programming: a tutorial. J. Comput. Appl. Math. 217, 394–419 (2008)

    Article  Google Scholar 

  13. Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Res. Natl. Bur. Stand. 49, 263–265 (1952)

    Article  Google Scholar 

  14. Khanh, P.Q., Luc, D.T.: Stability of solution sets in variational relation problems. Set Valued Anal. 16, 1015–1035 (2008)

    Article  Google Scholar 

  15. Lin, L.J., Ansari, Q.H.: Systems of quasi-variational relations with applications. Nonlinear Anal. 72, 1210–1220 (2010)

    Article  Google Scholar 

  16. Luc, D.T.: An abstract problem in variational analysis. J. Optim. Theory Appl. 138, 65–76 (2008)

    Article  Google Scholar 

  17. Luc, D.T., Sarabi, E., Soubeyran, A.: Existence of solutions in variational relation problems without convexity. J. Math. Anal. Appl. 364, 544–555 (2010)

    Article  Google Scholar 

  18. Mordukhovich, B.: Variational Analysis and Generalized Differentiation. I. Basic Theory and II: Applications. Springer, Berlin (2006)

    Google Scholar 

  19. Noor, M.A.: Modified resolvent algorithm for general mixed quasi-variational inequalities. Math. Comput. Model. 36, 737–745 (2002)

    Article  Google Scholar 

  20. Noor, M.A., Noor, K.I., Al-Said, E.: Iterative methods for solving general quasi-variational inequalities. Optim. Lett. 4, 513–530 (2010)

    Article  Google Scholar 

  21. Pu, Y.J., Yang, Z.: Stability of solutions for variational relation problems with applications. Nonlinear Anal. 75, 1758–1767 (2012)

    Article  Google Scholar 

  22. Pu, Y.J., Yang, Z.: Variational relation problems without the KKM property with applications. J. Math. Anal. Appl. 393, 256–264 (2012)

    Article  Google Scholar 

  23. Sach, P.H., Lin, L.J., Tuan, L.A.: Generalized vector quasivariational inclusion problems with moving cones. J. Optim. Theory Appl. 147, 607–620 (2010)

    Article  Google Scholar 

  24. Sach, P.H., Tuan, L.A., Minh, N.B.: Approximate duality for vector quasi-equilibrium problems and applications. Nonlinear Anal. 72, 3994–4004 (2010)

    Article  Google Scholar 

  25. 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. 56, 373–397 (2013)

    Google Scholar 

Download references

Acknowledgments

Anulekha Dhara would like to thank Université d’Avignon et des Pays du Vaucluse, Avignon, France for providing the postdoctoral fellowship during her visit when this work was carried out.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anulekha Dhara.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dhara, A., Luc, D.T. A solution method for linear variational relation problems. J Glob Optim 59, 729–756 (2014). https://doi.org/10.1007/s10898-013-0095-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-013-0095-5

Keywords

Navigation