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.
Similar content being viewed by others
References
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)
Agarwal, R.P., Balaj, M., O’Regan, D.: A unifying approach to variational relation problems. J. Optim. Theory Appl. 155, 417–429 (2012)
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)
Balaj, M.: A fixed point-equilibrium theorem with applications. Bull. Belg. Math. Soc. Simon Stevin 17, 919–928 (2010)
Balaj, M., Lin, L.J.: Equivalent forms of a generalized KKM theorem and their applications. Nonlinear Anal. 73, 673–682 (2010)
Balaj, M., Lin, L.J.: Generalized variational relation problems with applications. J. Optim. Theory Appl. 148, 1–13 (2011)
Balaj, M., Luc, D.T.: On mixed variational relation problems. Comput. Math. Appl. 60, 2712–2722 (2010)
Barber, C.B., Dobkin, D.P., Huhdanpaa, H.T.: The Quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22, 469–483 (1996)
Ding, X.P.: Algorithm of solutions for mixed implicit quasi-variational inequalities with fuzzy mappings. Comput. Math. Appl. 38, 231–241 (1999)
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)
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)
Guerra Vázqueza, F., Ruckmann, J.-J., Stein, O., Still, G.: Generalized semi-infinite programming: a tutorial. J. Comput. Appl. Math. 217, 394–419 (2008)
Hoffman, A.J.: On approximate solutions of systems of linear inequalities. J. Res. Natl. Bur. Stand. 49, 263–265 (1952)
Khanh, P.Q., Luc, D.T.: Stability of solution sets in variational relation problems. Set Valued Anal. 16, 1015–1035 (2008)
Lin, L.J., Ansari, Q.H.: Systems of quasi-variational relations with applications. Nonlinear Anal. 72, 1210–1220 (2010)
Luc, D.T.: An abstract problem in variational analysis. J. Optim. Theory Appl. 138, 65–76 (2008)
Luc, D.T., Sarabi, E., Soubeyran, A.: Existence of solutions in variational relation problems without convexity. J. Math. Anal. Appl. 364, 544–555 (2010)
Mordukhovich, B.: Variational Analysis and Generalized Differentiation. I. Basic Theory and II: Applications. Springer, Berlin (2006)
Noor, M.A.: Modified resolvent algorithm for general mixed quasi-variational inequalities. Math. Comput. Model. 36, 737–745 (2002)
Noor, M.A., Noor, K.I., Al-Said, E.: Iterative methods for solving general quasi-variational inequalities. Optim. Lett. 4, 513–530 (2010)
Pu, Y.J., Yang, Z.: Stability of solutions for variational relation problems with applications. Nonlinear Anal. 75, 1758–1767 (2012)
Pu, Y.J., Yang, Z.: Variational relation problems without the KKM property with applications. J. Math. Anal. Appl. 393, 256–264 (2012)
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)
Sach, P.H., Tuan, L.A., Minh, N.B.: Approximate duality for vector quasi-equilibrium problems and applications. Nonlinear Anal. 72, 3994–4004 (2010)
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)
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
Corresponding author
Rights 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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-013-0095-5