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.
Original language | English |
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Pages (from-to) | 1274-1285 |
Number of pages | 12 |
Journal | SIAM Journal on Optimization |
Volume | 20 |
Issue number | 3 |
DOIs | |
State | Published - 1 Dec 2009 |
Externally published | Yes |
Keywords
- Aubin property
- Lipschitz-like property
- Metric regularity
- Normal operator
- Parametric variational systems
- Quasi-monotone map
- Single-directional property