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Fan, J. H.; Wang, X. G.

Solvability of generalized variational inequality problems for unbounded sets in reflexive Banach spaces. (English) Zbl 1179.58009

J. Optim. Theory Appl. 143, No. 1, 59-74 (2009).
Summary: By employing the notion of exceptional family of elements, we establish some existence results for generalized variational inequality problems in reflexive Banach spaces provided that the mapping is upper sign-continuous. We show that the nonexistence of an exceptional family of elements is a necessary condition for the solvability of the dual variational inequality. For quasimonotone variational inequalities, we present some sufficient conditions for the existence of strong solutions. For the pseudomonotone case, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the problem having strong solutions. Furthermore, we establish several equivalent conditions for the solvability in the pseudomonotone case. As a byproduct, a quasimonotone generalized variational inequality is proved to have a strong solution if it is strictly feasible. Moreover, for the pseudomonotone case, the strong solution set is nonempty and bounded if it is strictly feasible.

Cited in 5 Documents

MSC:

58E35 Variational inequalities (global problems) in infinite-dimensional spaces

Keywords:

generalized variational inequality problems; exceptional family of elements; existence of solutions; pseudomonotone mappings; quasimonotone mappings
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References:

[1] Smith, T.E.: A solution condition for complementarity problems, with an application to spatial price equilibrium. Appl. Math. Comput. 15, 61–69 (1984) · Zbl 0545.90094 · doi:10.1016/0096-3003(84)90053-5
[2] Isac, G., Kalashnikov, V.: Exceptional family of elements, Leray-Schauder alternative, pseudomonotone operators, and complementarity. J. Optim. Theory Appl. 109, 69–83 (2001) · Zbl 0984.49006 · doi:10.1023/A:1017509704362
[3] Isac, G., Li, J.: Exceptional family of elements and the solvability of complementarity problems in uniformly smooth and uniformly convex Banach spaces. J. Zhejiang Univ. Sci. 6A, 289–295 (2005) · Zbl 1107.47039 · doi:10.1631/jzus.2005.A0289
[4] Li, J., Huang, N.J.: Exceptional families of elements for set-valued mappings: an application to nonlinear complementarity problems. Appl. Math. Lett. 21, 42–46 (2008) · Zbl 1140.47037 · doi:10.1016/j.aml.2006.07.017
[5] Zhao, Y.B., Yuan, J.Y.: An alternative theorem for generalized variational inequalities and solvability of nonlinear quasi-P * M -complementarity problems. Appl. Math. Comput. 109, 167–182 (2000) · Zbl 1021.49009 · doi:10.1016/S0096-3003(99)00019-3
[6] Zhao, Y.B.: Exceptional family and finite-dimensional variational inequality over polyhedral convex set. Appl. Math. Comput. 87, 111–126 (1997) · Zbl 0912.49009 · doi:10.1016/S0096-3003(96)00224-X
[7] Han, J., Huang, Z.H., Fang, S.C.: Solvability of variational inequality problems. J. Optim. Theory Appl. 122, 501–520 (2004) · Zbl 1082.49009 · doi:10.1023/B:JOTA.0000042593.74934.b7
[8] Bianchi, M., Hadjisavvas, N., Schaible, S.: Exceptional families of elements for variational inequalities in Banach spaces. J. Optim. Theory Appl. 129, 23–31 (2006) · Zbl 1136.49007 · doi:10.1007/s10957-006-9041-8
[9] Isac, G., Zhao, Y.B.: Exceptional family of elements and the solvability of variational inequalities for unbounded sets in infinite-dimensional Hilbert spaces. J. Math. Anal. Appl. 246, 544–556 (2000) · Zbl 0966.49006 · doi:10.1006/jmaa.2000.6817
[10] Li, J., Whitaker, J.: Exceptional family of elements and solvability of variational inequalities for mappings defined only on closed convex cones in Banach spaces. J. Math. Anal. Appl. 310, 254–261 (2005) · Zbl 1135.49008 · doi:10.1016/j.jmaa.2005.02.006
[11] Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I, II. Springer, Berlin (2003) · Zbl 1062.90002
[12] He, Y.R.: Stable pseudomonotone variational inequality in reflexive Banach spaces. J. Math. Anal. Appl. 330, 352–363 (2007) · Zbl 1124.49005 · doi:10.1016/j.jmaa.2006.07.063
[13] He, Y.R., Miao, X.Z., Zhou, M.: Strict feasibility of variational inequalities in reflexive Banach spaces. Acta Math. Sinica 23, 563–570 (2007) · Zbl 1126.47049 · doi:10.1007/s10114-005-0918-5
[14] Zhao, Y.B., Li, D.: Strict feasibility conditions in nonlinear complementarity problems. J. Optim. Theory Appl. 107, 641–664 (2000) · Zbl 1168.90625 · doi:10.1023/A:1026459501988
[15] Aussel, D., Luc, D.T.: Existence conditions in general quasimonotone variational inequality. Bull. Aust. Math. Soc. 71, 285–303 (2005) · Zbl 1095.49007 · doi:10.1017/S0004972700038259
[16] Bianchi, M., Pini, R.: Coercivity conditions for equilibrium problems. J. Optim. Theory Appl. 124, 79–92 (2005) · Zbl 1064.49004 · doi:10.1007/s10957-004-6466-9
[17] Flores Bazán, F.: Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case. SIAM J. Optim. 11, 675–690 (2000) · Zbl 1002.49013
[18] Crouzeix, J.P.: Pseudomonotone variational inequality problems: existence of solutions. Math. Program. 78, 305–314 (1997) · Zbl 0887.90167
[19] Daniilidis, A., Hadjisavvas, N.: Coercivity conditions and variational inequalities. Math. Program. 86, 433–438 (1999) · Zbl 0937.49003 · doi:10.1007/s101070050097
[20] Hadjisavvas, N.: Continuity and maximality properties of pseudomonotone operators. J. Convex Anal. 20, 465–475 (2003) · Zbl 1063.47041
[21] Aubin, J.P.: Optima and Equilibria. Springer, Berlin (1993) · Zbl 0781.90012
[22] Bianchi, M., Hadjisavvas, N., Schaible, S.: Minimal coercivity conditions and exceptional families of elements in quasimonotone variational inequalities. J. Optim. Theory Appl. 122, 1–17 (2004) · Zbl 1130.49302 · doi:10.1023/B:JOTA.0000041728.12683.89
[23] Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990) · Zbl 0734.90098 · doi:10.1007/BF01582255
[24] Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984) · Zbl 0641.47066
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