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Vector variational-like inequalities with generalized bifunctions defined on nonconvex sets. (English) Zbl 1182.47046
In this paper, the nonemptiness and compactness of solution sets for Stampacchia vector variational-like inequalities (for short, SVVLIs) and Minty vector variational-like inequalities (for short, MVVLIs) with generalized bifunctions defined on nonconvex sets are investigated by introducing the concepts of generalized weak cone-pseudomonotonicity and generalized (proper) cone-suboddness. The equivalent relations between a solution of SVVLIs and MVVLIs, and a generalized weakly efficient solution of vector optimization problems are established under the assumptions of generalized pseudoconvexity and generalized invexity in the sense of the Clarke generalized directional derivative.

##### MSC:
 47J20 Variational and other types of inequalities involving nonlinear operators (general) 90C29 Multi-objective and goal programming
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##### References:
 [1] Giannessi, F., Theorem of alternative quadratic programs, and complementarity problems, (), 151-186 [2] Ceng, L.C.; Schaible, S.; Yao, J.C., A generalization of minty’s lemma and its applications to generalized mixed vector equilibrium problems, Optimization, 56, 1-12, (2007) · Zbl 1128.49014 [3] Chen, G.Y., Existence of solutions for a vector variational inequality: an extension of hartmann – stampacchia theorem, J. optim. theory appl., 74, 445-456, (1992) · Zbl 0795.49010 [4] Chen, G.Y.; Huang, X.X.; Yang, X.Q., () [5] Giannessi, F., On minty variational principle, (), 93-99 · Zbl 0909.90253 [6] () [7] Kim, M.H.; Lee, G.M., Some existence results for vector optimization problems, (), 147-157 [8] Kimura, K.; Yao, J.C., Sensitivity analysis of vector equilibrium problems, Taiwanese J. math., 12, 649-669, (2008) · Zbl 1159.49025 [9] Konnov, I.V.; Yao, J.C., On the generalized vector variational inequality problem, J. math. anal. appl., 206, 42-58, (1997) · Zbl 0878.49006 [10] Lalitha, C.S.; Mehta, M., Vector variational inequalities with cone-pseudomonotone bifunctions, Optimization, 54, 327-338, (2005) · Zbl 1087.90069 [11] Luc, D.T., () [12] Ward, D.E.; Lee, G.M., On relations between vector optimization problems and vector variational inequalities, J. optim. theory appl., 113, 583-596, (2002) · Zbl 1022.90024 [13] Yang, X.M.; Yang, X.Q.; Teo, K.L., Some remarks on the minty vector variational inequality, J. optim. theory appl., 121, 193-201, (2004) · Zbl 1140.90492 [14] Zeng, L.C.; WU, S.Y.; Yao, J.C., Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems, Taiwanese J. math., 10, 1497-1514, (2006) · Zbl 1121.49005 [15] Ceng, L.C.; Chen, G.Y.; Huang, X.X.; Yao, J.C., Existence theorems for generalized vector variational inequalities with pseudomonotonicity and their applications, Taiwanese J. math., 12, 151-172, (2008) · Zbl 1148.49004 [16] Crouzeix, J.P., Pseudomonotone variational inequality problems: existence of solutions, Math. program. ser. A, 78, 305-314, (1997) · Zbl 0887.90167 [17] Harker, P.T.; Pang, J.S., Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory algorithms and applications, Math. program. ser. B, 48, 161-220, (1990) · Zbl 0734.90098 [18] Huang, N.J.; Li, J.; Thompson, H.B., Implicit vector equilibrium problems with applications, Math. comput. modelling, 37, 1343-1356, (2003) · Zbl 1080.90086 [19] Kien, B.T.; Wong, N.C.; Yao, J.C., Generalized vector variational inequalities with star-pseudomonotone and discontinuous operators, Nonlinear anal. ser. A: TMA, 68, 2859-2871, (2008) · Zbl 1336.49010 [20] Yao, J.C., Multi-valued variational inequalities with $$K$$-pseudomonotone operators, J. optim. theory appl., 83, 391-403, (1994) · Zbl 0812.47055 [21] Yao, J.C., Variational inequalities with generalized monotone operators, Math. oper. res., 19, 691-705, (1994) · Zbl 0813.49010 [22] Mangasarian, O.L., Pseudo-convex functions, SIAM J. control, 3, 281-290, (1965) · Zbl 0138.15702 [23] Komlósi, S., Generalized monotonicity and generalized convexity, J. optim. theory appl., 84, 361-376, (1995) · Zbl 0824.90124 [24] Aubin, J.P., Optima and equilibria: an introduction to nonlinear analysis, (1998), Springer-Verlag Berlin, (Translated by Stephen Wilson) [25] Rockafellar, R.T., Convex analysis, (1970), Princeton University Press Princeton, NJ · Zbl 0229.90020 [26] Sach, P.H.; Penot, J.P., Characterizations of generalized convexities via generalized directional derivatives, Numer. funct. anal. optim., 19, 615-634, (1998) · Zbl 0916.49015 [27] Fan, K., A generalization of tychonoff’s fixed point theorem, Math. ann., 142, 305-310, (1961) · Zbl 0093.36701 [28] Yang, X.Q., On the gap functions of prevariational inequalities, J. optim. theory appl., 116, 437-452, (2003) · Zbl 1027.49004 [29] Clarke, F.H., Optimization and nonsmooth analysis, (1983), Wiley-Interscience New York · Zbl 0727.90045 [30] Giorgi, G.; Guerraggio, A., Various types of nonsmooth invex functions, J. inf. optim. sci., 17, 137-150, (1996) · Zbl 0859.49020 [31] Hanson, M.A., On sufficiency of the kuhn – tucker conditions, J. math. anal. appl., 80, 545-550, (1981) · Zbl 0463.90080
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