×

Stability of weak vector variational inequality. (English) Zbl 1158.49018

Summary: A key assumption is introduced by virtue of a parametric gap function. Then, by using the key assumption, sufficient conditions of the continuity and Hausdorff continuity of a solution set map for a parametric weak vector variational inequality are obtained in Banach spaces with the objective space being finite-dimensional.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J40 Variational inequalities
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anh, L. Q.; Khanh, P. Q., Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems, Journal of Mathematical Analysis and Applications, 294, 699-711 (2004) · Zbl 1048.49004
[2] Bank, B.; Guddat, J.; Klatte, D.; Kummer, B.; Tammer, K., Non-Linear Parametric Optimization (1982), Akademie-Verlag: Akademie-Verlag Berlin · Zbl 0502.49002
[3] Chen, G. Y., Existence of solutions for a vector variational inequality: An extension of the Hartmann-Stampacchia theorem, Journal of Optimization Theory and Applications, 74, 445-456 (1992) · Zbl 0795.49010
[4] Chen, G. Y.; Li, S. J., Existence of solutions for a generalized vector quasivariational inequality, Journal of Optimization Theory and Applications, 90, 321-334 (1996) · Zbl 0869.49005
[5] Chen, G. Y.; Goh, C. J.; Yang, X. Q., On gap functions for vector variational inequalities, (Giannessi, F., Vector Variational Inequalities and Vector Equilibria (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Holland), 55-72 · Zbl 0997.49006
[6] Chen, G. Y.; Huang, X. X.; Yang, X. Q., (Vector Optimization: Set-Valued and Variational Analysis. Vector Optimization: Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems, 541 (2005), Springer: Springer Berlin) · Zbl 1104.90044
[7] Cheng, Y. H.; Zhu, D. L., Global stability results for the weak vector variational inequality, Journal of Global Optimization, 32, 543-550 (2005) · Zbl 1097.49006
[8] Chiang, Y., Semicontinuous mappings into T.V.S. with applications to mixed vector variational-like inequalities, Journal of Global Optimization, 32, 467-484 (2005) · Zbl 1100.49008
[9] Giannessi, F., Theorems of the alternative, quadratic programs and complementary problems, (Cottle, R. W.; Giannessi, F.; Lions, J. L., Variational Inequalities and Complementary Problems (1980), Wiley: Wiley New York), 151-186 · Zbl 0484.90081
[10] (Giannessi, F., Vector Variational Inequalities and Vector Equilibria: Mathematical Theories (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Holland) · Zbl 0952.00009
[11] Khanh, P. Q.; Luu, L. M., Upper semicontinuity of the solution set to parametric vector quasivariational inequalities, Journal of Global Optimization, 32, 569-580 (2005) · Zbl 1097.49013
[12] Kien, B. T., On the lower semicontinuity of optimal solution sets, Optimization, 54, 123-130 (2005) · Zbl 1141.90551
[13] Kimura, K.; Yao, J. C., Sensitivity analysis of vector equilibrium problems, Taiwanese Journal of Mathematics, 12, June. (2008) · Zbl 1159.49025
[14] Kimura, K.; Yao, J. C., Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems, Journal of Industrial Management and Optimization, 4, 167-181 (2008) · Zbl 1158.49017
[15] Li, S. J.; Chen, G. Y.; Teo, K. L., On the stability of generalized vector quasivariational inequality problems, Journal of Optimization Theory and Applications, 113, 283-295 (2002) · Zbl 1003.47049
[16] Li, S. J.; Teo, K. L.; Yang, X. Q.; Wu, S. Y., Gap functions and existence of solutions to generalized vector quasi-equilibrium problems, Journal of Global Optimization, 34, 427-440 (2006) · Zbl 1090.49014
[17] Yang, X. Q.; Yao, J. C., Gap functions and existence of solutions to set-valued vector variational inequalities, Journal of Optimization Theory and Applications, 115, 407-417 (2002) · Zbl 1027.49003
[18] Yang, X. Q.; Yu, H., Vector variational inequalities and dynamic traffic equilibria, (Giannessi, F.; Maugeri, A., Variational Analysis and Applications (2005), Springer), 1141-1157 · Zbl 1113.49014
[19] Zhao, J., The lower semicontinuity of optimal solution sets, Journal of Mathematical Analysis and Applications, 207, 240-254 (1997) · Zbl 0872.90093
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.