DOI QR코드

DOI QR Code

SCALARIZATION METHODS FOR MINTY-TYPE VECTOR VARIATIONAL INEQUALITIES

  • Received : 2009.12.31
  • Accepted : 2010.05.01
  • Published : 2010.05.31

Abstract

Many kinds of Minty's lemmas show that Minty-type variational inequality problems are very closely related to Stampacchia-type variational inequality problems. Particularly, Minty-type vector variational inequality problems are deeply connected with vector optimization problems. Liu et al. [10] considered vector variational inequalities for setvalued mappings by using scalarization approaches considered by Konnov [8]. Lee et al. [9] considered two kinds of Stampacchia-type vector variational inequalities by using four kinds of Stampacchia-type scalar variational inequalities and obtain the relations of the solution sets between the six variational inequalities, which are more generalized results than those considered in [10]. In this paper, the author considers the Minty-type case corresponding to the Stampacchia-type case considered in [9].

Acknowledgement

Supported by : Kyungsung University

References

  1. G. Y. Chen, X. X. Huang and X. Q. Yang, Vector optimization: Set-valued Variational Analysis, Springer-Verlag, Berlin, Heidelberg, 2005.
  2. F. Giannessi, Vector Variational Inequalities and Vector Theory Variational Equilib- rium, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000.
  3. F. Giannessi, G. Mastroeni and L. Pellegrini, On the theory of vector optimization and variational inequalities. Image space analysis and separation, in : F. Giannessi(Ed.) Vector variational Inequalities and Vector Equilibria, Kluwer Academic Publishers, Dor- drecht, Boston, London, 2000, pp.153-215.
  4. S.-M. Guu, N.-J. Huang and J. Li, Scalarization approaches for set-valued vector opti-mization problems and vector variational inequalities, J. Math. Anal. Appl. 356 (2009), 564-576. https://doi.org/10.1016/j.jmaa.2009.03.040
  5. N. Hadjesavvas and S. Schaible, Quasimonotonicity and pseudomonotonicity in vari- ational inequalities and equilibrium problem: In Generalized Convexity Generalized Monotonicity (Edited by J.P. Crouzeix, J.E. Martinez-Legaz and M. Volle) Academic Publishers, Dordrecht, 257-275, 1998.
  6. B. Jimenez, V. Novo and M. Sama, Scalarization and optimality conditions for strict minimizers in multiobjective optimization via contingent epiderivatives, J. Math. Anal. Appl. 352 (2009), 788-798. https://doi.org/10.1016/j.jmaa.2008.11.045
  7. H. Kneser, Sur un theoreme fundamental dela theorie des Jeux, C. R. Acad. Sci. Paris 234 (1952), 2418-2420.
  8. I. V. Konnov, A scalarization approach for variational inequalities with applications, J. Global Optim. 32 (2005), 517-527. https://doi.org/10.1007/s10898-003-2688-x
  9. B.-S. Lee, M. Firdosh Khan and Salahuddin, Scalarization methods for vector variational inequalities, to be appeared.
  10. Z. B. Liu, N. J. Huang and B. S. Lee, Scalarization approaches for generalized vector variational inequalities, Nonlinear Anal. Forum 12(1) (2007), 119-124.
  11. Z. D. Slavov, Scalarization techniques or relationship between a social welfare function and a Pareto optimality concept, Appl. Math. & Comp. 172 (2006), 464-471. https://doi.org/10.1016/j.amc.2005.02.013