Lower semicontinuity of the solution map to a parametric generalized variational inequality in reflexive Banach spaces. (English) Zbl 1210.47088

In this paper, the authors discuss the stability of the solution set for the following parametric generalized variational inequality, denoted by \(GVI(F(\mu,\cdot),K(\lambda))\), which consists in finding \(x(\mu,\lambda)\) such that
\[ 0 \in F(\mu,x)+N_{K(\lambda)}(x), \]
where \(E\) is a normed space with the dual \(E^*\), \((M, d)\) and \((\Lambda, d)\) are metric spaces, \(F : M \times E \rightarrow 2^{E^*}\) is a set-valued operator, and \(K: \Lambda\rightarrow 2^E\) is a multifunction with closed convex values.
They prove that the solution map of \(GVI(F(\mu,\cdot),K(\lambda))\) is lower semicontinuous when the operator of the unperturbed problem is of the class \((S)_+\) and strictly monotone, the operators of the perturbed problems are pseudo-monotone and demicontinuous, and the perturbed constraint set \(K(\cdot)\) has the Aubin property. The obtained results are proved without conditions related to degree theory and the metric projection. Thus, they do not have to verify the conditions via the metric projection and computing of the degree of the normal map which were required by B.T.Kien and J.-C.Yao [Set-Valued Anal.16, No.4, 399–412 (2008; Zbl 1210.47089)].


47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J40 Variational inequalities
49J53 Set-valued and variational analysis
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)


Zbl 1210.47089
Full Text: DOI


[1] Bessis, D.N., Ledyaev, Yu.S.,Vinter, R.B.: Dualization of the Euler and Hamiltonian inclusions. Nonlinear Anal. 43, 861–882 (2001) · Zbl 1004.49016
[2] Browder, F.E., Hess, P.: Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal. 11, 251–294 (1972) · Zbl 0249.47044
[3] Chang, D., Pang, J.S.: The generalized quasi-variational inequality problem. Math. Oper. Res. 2, 211–222 (1982) · Zbl 0502.90080
[4] Cioranescu, I.: Geometry of Banach Spaces Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht (1990) · Zbl 0712.47043
[5] Dafermos, S.: Sensitivity analysis in variational inequalities. Math. Oper. Res. 13, 421–434 (1988) · Zbl 0674.49007
[6] Domokos, A.: Solution sensitivity of variational inequalities. J. Math. Math. Appl. 230, 382–389 (1999) · Zbl 0927.49005
[7] Dontchev, A.L., Hager, W.W.: Impicit functions, Lipschitz maps, and stability in optimization. Math. Oper. Res. 19, 753–768 (1994) · Zbl 0835.49019
[8] Dontchev, A.L.: Implicit function theorems for generalized equations. Math. Programming 70, 91–106 (1995) · Zbl 0843.49010
[9] Kien, B.T., Wong, M.M., Wong, N.C., Yao, J.C.: Solution existence of variational inequalities with pseudomonotone operators in the sense of Brezis. J. Optim. Theory Appl. (2008, in press) · Zbl 1162.58006
[10] Kien, B.T., Yao, J.C.: Localization of generalized normal maps and stability of variational inequalities in reflexive Banach spaces. Set-Valued Anal. (2008, in press) · Zbl 1210.47089
[11] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications. Academic, London (1980) · Zbl 0457.35001
[12] Levy, A.B., Rockafellar, R.T.: Sensitivity analysis of solutions to generalized equations. Trans. Amer. Math. Soc. 345, 661–671 (1994) · Zbl 0815.47077
[13] Levy, A.B.: Sensitivity of solutions to variational inequalities on Banach spaces. SIAM J. Control Optim. 38, 50–60 (1999) · Zbl 0951.49031
[14] Levy, A.B., Mordukhovich, B.S.: Coderivatives in parametric optimization. Math. Programming 99, 311–327 (2004) · Zbl 1079.90136
[15] Mangasarian, O.L., Shiau, T.-H.: Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems. SIAM J Control Optim. 25, 583–595 (1987) · Zbl 0613.90066
[16] Mansour, M.A., Aussel, D.: Quasimonotone variational inequalities and qusiconvex programming: qualitatve stability. Pac. J. Optim. 2, 611–626 (2006) · Zbl 1124.49023
[17] Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, New York (2006)
[18] Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, New York (2006)
[19] Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 130–143 (1976) · Zbl 0418.52005
[20] Robinson, S.M.: An implicit-function theorem for a class of nonsmooth functions. Math. Oper. Res. 16, 292–309 (1991) · Zbl 0746.46039
[21] Robinson, S.M.: Normal maps induced by linear transformations. Math. Oper. Res. 17, 691–714 (1992) · Zbl 0777.90063
[22] Robinson, S.M.: Constraint nondegeneracy in variational analysis. Math. Oper. Res. 28, 201–232 (2003) · Zbl 1082.90116
[23] Robinson, S.M.: Localized normal maps and the stability of variational conditions. Set-Valued Anal. 12, 259–274 (2004) · Zbl 1066.47064
[24] Robinson, S.M.: Solution continuity affine variational inequalities. SIAM. J. Optim. 18, 1046–1060 (2007) · Zbl 1143.49015
[25] Robinson, S.M., Lu, S.: Solution continuity in variational conditions. J. Glob. Optim. (2008, in press) · Zbl 1145.90096
[26] Rockafellar, R.T., Wets, R.J.: Variational Analysis. Springer, Berlin (1998)
[27] Sion, M.: On general minimax theorems. Pacific J. Math. 8, 171–176 (1958) · Zbl 0081.11502
[28] Yen, N.D.: Hölder continuity of solution to a parametric variational inequality. Appl. Math. Optim. 31, 245–255 (1995) · Zbl 0821.49011
[29] Yen, N.D.: Lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint. Math. Oper. Res. 20, 695–707 (1995) · Zbl 0845.90116
[30] Yen, N.D., Lee, G.M.: Solution sensitivity of a class of variational inequalities. J. Math. Anal. Appl. 215, 48–55 (1997) · Zbl 0906.49002
[31] Zeidler, E.: Nonlinear Functional Analysis and its Application, II/B: Nonlinear Monotone Operators. Springer, Heidelberg (1990) · Zbl 0684.47029
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