×

Existence of solutions for vector optimization problems. (English) Zbl 0911.90290

Summary: We define the generalized efficient solution which is more general than the weakly efficient solution for vector optimization problems, and prove the existence of the generalized efficient solution for nondifferentiable vector optimization problems by using vector variational-like inequalities for set-valued maps. \(\copyright\) Academic Press.

MSC:

90C29 Multi-objective and goal programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Borwein, J. M., On the existence of Pareto efficient points, Math. Oper. Res., 8, 64-73 (1983) · Zbl 0508.90080
[2] Corley, H. W., An existence result for maximization with respect to cones, J. Optim. Theory Appl., 31, 277-281 (1980) · Zbl 0416.90068
[3] Jahn, J., Mathematical Vector Optimization in Partially Ordered Linear Spaces (1986), Verlag Peter Lang · Zbl 0578.90048
[4] Yu, P. L., Cone convexity, cone extreme points and nondominated solutions in decision problems with multi-objectives, J. Optim. Theory Appl., 14, 319-377 (1974) · Zbl 0268.90057
[5] Chen, G. Y.; Craven, B. D., Existence and continuity of solutions for vector optimization, J. Optim. Theory Appl., 81, 459-468 (1994) · Zbl 0810.90112
[6] Fan, K., A generalization of Tychonoff’s fixed point theorem, Math. Ann., 142, 305-310 (1961) · Zbl 0093.36701
[7] Kazmi, K. R., Existence of solutions for vector optimization, Appl. Math. Lett., 9, 19-22 (1996) · Zbl 0869.90063
[8] Giannessi, F., Theorems of alternative, quadratic programs and complementarity problems, (Cottle, R. W.; Giannessi, F.; Lions, J. L., Variational Inequalities and Complementarity Problems (1980), Wiley: Wiley Chichester) · Zbl 0484.90081
[9] Chen, G. Y., Existence of solutions for a vector variational inequality: An extension of the Hartman-Stampacchia theorem, J. Optim. Theory Appl., 74, 445-456 (1992) · Zbl 0795.49010
[10] Chen, G. Y.; Li, S. J., Existence of solutions for generalized vector quasi-variational inequality, J. Optim. Theory Appl., 90, 321-334 (1996) · Zbl 0869.49005
[11] Konnov, I. V.; Yao, J. C., On the generalized vector variational inequality problem, J. Math. Anal. Appl., 206, 42-58 (1997) · Zbl 0878.49006
[12] Lee, B. S.; Lee, G. M.; Kim, D. S., Generalized vector variational-like inequalities on locally convex Hausdorff topological vector spaces, Indian J. Pure Appl. Math., 28, 33-41 (1997) · Zbl 0899.49005
[13] Lai, T. C.; Yao, J. C., Existence results for VVIP, Appl. Math. Lett., 9, 17-19 (1996) · Zbl 0886.49008
[14] Lee, G. M.; Kim, D. S.; Lee, B. S.; Cho, S. J., Generalized vector variational inequality and fuzzy extension, Appl. Math. Lett., 6, 47-51 (1993) · Zbl 0804.49004
[15] Lee, G. M.; Kim, D. S.; Lee, B. S.; Cho, S. J., On vector variational inequality, Bull. Korean Math. Soc., 33, 553-564 (1996) · Zbl 0871.49011
[16] Lee, G. M.; Lee, B. S.; Chang, S. S., On vector quasivariational inequalities, J. Math. Anal. Appl., 203, 626-638 (1996) · Zbl 0866.49016
[17] G. M. Lee, D. S. Kim, B. S. Lee, N. D. Yen, Vector variational inequality as a tool for studying vector optimization problems, Nonlinear Anal.; G. M. Lee, D. S. Kim, B. S. Lee, N. D. Yen, Vector variational inequality as a tool for studying vector optimization problems, Nonlinear Anal. · Zbl 1018.90053
[18] Lin, L. J., Pre-vector variational inequalities, Bull. Austral. Math. Soc., 53, 63-70 (1996) · Zbl 0858.49008
[19] Lin, K. L.; Yang, D. P.; Yao, J. C., Generalized vector variational inequalities, J. Optim. Theory Appl., 92, 117-125 (1997) · Zbl 0886.90157
[20] Siddiqi, A. H.; Ansari, A. H.; Khaliq, A., On vector variational inequalities, J. Optim. Theory Appl., 84, 171-180 (1995) · Zbl 0827.47050
[21] Yang, X. Q., Generalized convex functions and vector variational inequalities, J. Optim. Theory Appl., 79, 563-579 (1993) · Zbl 0797.90085
[22] Yu, S. J.; Yao, J. C., On vector variational inequalities, J. Optim. Theory Appl., 89, 749-769 (1996) · Zbl 0848.49012
[23] Clarke, F. H., Optimization and Nonsmooth Analysis (1983), Wiley-Interscience: Wiley-Interscience New York · Zbl 0727.90045
[24] Hanson, M. A., On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80, 545-550 (1982) · Zbl 0463.90080
[25] Giorgi, G.; Guerraggio, A., Various types of nonsmooth invex functions, J. Inform. Optim. Sci., 17, 137-150 (1996) · Zbl 0859.49020
[26] Sawaragi, Y.; Nakayama, H.; Tanino, T., Theory of Multiobjective Optimization (1985), Academic Press: Academic Press San Diego · Zbl 0566.90053
[27] Yao, J. C.; Guo, J. S., Variational and generalized variational inequalities with discontinuous mappings, J. Math. Anal. Appl., 182, 371-392 (1994) · Zbl 0809.49005
[28] Aubin, J. P., Optima and Equilibria (1993), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York
[29] Chang, S. S.; Lee, B. S.; Wu, X.; Cho, Y. J.; Lee, G. M., On the generalized quasi-variational inequality problems, J. Math. Anal. Appl., 203, 686-711 (1996) · Zbl 0867.49008
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.