Single-Directional Property of Multivalued Maps and Variational Systems
Abstract
Dontchev and Hager [Math. Oper. Res., 19 (1994), pp. 753–768] have shown that a monotone set-valued map defined from a Banach space to its dual which satisfies the Aubin property around a point $(x,y)$ of its graph is actually single-valued in a neighborhood of x. We prove a result which is the counterpart of the above for quasi-monotone set-valued maps, based on the concept of single-directional property. As applications, we provide sufficient conditions for this single-valued property to hold for the solution map of general variational systems and quasi-variational inequalities. We also investigate the single-directionality property for the normal operator to the sublevel sets of a quasi-convex function.
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References
1.
J.-P. Aubin and H. Frankowska, Set-valued analysis, Birkhäuser Boston, Boston, MA, 1990.
2.
F. J. Aragón Artacho and B. Mordukhovich, Metric regularity and Lipschitzian stability of parametric variational systems, Nonlinear Anal., to appear.
3.
D. Aussel and N. Hadjisavvas, Adjusted sublevel sets, normal operator, and quasi-convex programming, SIAM J. Optim., 16 (2005), pp. 358–367.
4.
D. Aussel and J. Ye, Quasi-Convex programming with starshaped constraint region and application to quasi-convex MPEC, Optimization, 55 (2006), pp. 433–457.
5.
J. Borde and J.-P. Crouzeix, Continuity properties of the normal cone to the level sets of a quasiconvex function, J. Optim. Theory Appl., 66 (1990), pp. 415–429.
6.
A. L. Dontchev and W. Hager, Implicit functions, Lipschitz maps, and stability in optimization, Math. Oper. Res., 19 (1994), pp. 753–768.
7.
A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings, Springer Monogr. Math., Springer-Verlag, Berlin, 2009.
8.
W. Geremew, B. Mordukhovich, and N. N. Nam, Coderivative calculus and metric regularity for constraint and variational systems, Nonlinear Anal., 70 (2009), pp. 529–552.
9.
N. Hadjisavvas, Continuity and maximality properties of pseudomonotone operators, J. Convex Anal., 10 (2003), pp. 459–469.
10.
S. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Kluwer Academic Publishers, Dordrecht, 1997.
11.
P. Kenderov, Semi-continuity of set-valued mappings, Fund. Math., 88 (1975), pp. 61–69.
12.
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Grundlehren Math. Wiss. 330, Springer-Verlag, Berlin, 2006.
13.
B. Mordukhovich, Failure of metric regularity for major classes of variational systems, Nonlinear Anal., 69 (2008), pp. 918–924.
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Copyright © 2009 Society for Industrial and Applied Mathematics.
History
Submitted: 18 September 2008
Accepted: 11 May 2009
Published online: 1 October 2009
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