×

Generalized vector variational inequality and its duality for set-valued maps. (English) Zbl 0940.49008

Summary: We consider a Generalized Vector Variational Inequality (GVVI) for set-valued maps, give its dual form DVVI, and prove an equivalence between GVVI and DVVI.

MSC:

49J40 Variational inequalities
49J53 Set-valued and variational analysis
Full Text: DOI

References:

[1] Giannessi, F., Theorems of alternative, quadratic programs and complementarity problems, (Cottle, R. W.; Giannessi, F.; Lions, J. L., Variational Inequalities and Complementarity Problems (1980), John Wiley & Sons: John Wiley & Sons Chichester, England), 151-186 · Zbl 0484.90081
[2] Chen, G. Y.; Cheng, G. M., Vector variational inequality and vector optimizations, (Lecture Notes in Economics and Mathematical Systems, Volume 285 (1987), Springer-Verlag), 408-416
[3] Chen, G. Y.; Craven, B. D., Approximate dual and approximate vector variational inequality for multiobjective optimization, J. Austral. Math. Soc. (Series A), 47, 418-423 (1989) · Zbl 0693.90089
[4] Chen, G. Y.; Craven, B. D., A vector variational inequality and optimization over an efficient set, Zor-Meth. Models Op. Res., 34, 1-12 (1990) · Zbl 0693.90091
[5] Chen, G. Y.; Yang, X. Q., The vector complementarity problem and its equivalence with the weak minimal element in ordered sets, J. Math. Anal. Appl., 153, 136-158 (1990) · Zbl 0712.90083
[6] Chen, G. Y., Existence of solutions for a vector variational inequality: An extension of Hartman-Stampacchia theorem, J. Optim. Th. Appl., 74, 3, 445-456 (1992) · Zbl 0795.49010
[7] Chen, G. Y.; Li, S. J., Existence of solutions for a generalized vector quasivariational inequality, J. Optim. Th. Appl., 90, 2, 321-334 (1996) · Zbl 0869.49005
[8] Lee, G. M.; Kim, D. S.; Lee, B. S.; Cho, S. J., Generalized vector variational inequality and fuzzy extension, Appl. Math. Lett., 6, 6, 47-51 (1993) · Zbl 0804.49004
[9] Lee, G. M.; Lee, B. S.; Chang, S.-S., On vector quasivariational inequalities, J. Math. Anal. Appl., 203, 626-638 (1966) · Zbl 0866.49016
[10] Lee, G. M.; Kim, D. S.; Lee, B. S., Generalized vector variational inequality, Appl. Math. Lett., 9, 1, 39-42 (1996) · Zbl 0862.49014
[11] Siddiqi, A. H.; Ansari, Q. H.; Khaliq, A., On vector variational inequalities, J. Optim. Th. Appl., 84, 171-180 (1995) · Zbl 0827.47050
[12] Yang, X. Q., Generalized convex functions and vector variational inequalities, J. Optim. Th. Appl., 79, 563-580 (1993) · Zbl 0797.90085
[13] Yang, X. Q., Vector variational inequality and its duality, Nonlinear Analysis, T.M.A., 21, 869-877 (1993) · Zbl 0809.49009
[14] Yu, S. J.; Yao, J. C., On vector variational inequalities, J. Optim. Th. Appl., 89, 749-769 (1996) · Zbl 0848.49012
[15] Fan, K., A generalization of Tychonoff’s fixed point theorem, Math. Ann., 142, 305-310 (1961) · Zbl 0093.36701
[16] Sawaragi, Y.; Nakayama, H.; Tanino, T., Theory of Multiobjective Optimization (1985), Academic Press: Academic Press New York · Zbl 0566.90053
[17] Yang, X. Q., A Hahn-Banach theorem in ordered linear space and its applications, Optimization, 25, 1-9 (1992) · Zbl 0834.46006
[18] Mosco, U., Dual variational inequalities, J. Math. Anal. Appl., 40, 202-206 (1972) · Zbl 0262.49003
[19] Jameson, G., Ordered linear spaces, (Lecture Notes in Mathematics, Volume 141 (1970), Springer: Springer New York) · Zbl 0196.13401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.