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Zbl 0955.90138
Levy, A.B.
Stability of solutions to parameterized nonlinear complementarity problems. (English)
[J] Math. Program. 85, No.2 (A), 397-406 (1999). ISSN 0025-5610; ISSN 1436-4646/e

Summary: We consider the stability properties of solutions to parameterized nonlinear complementarity problems $$\text{Find }x\in \bbfR^n \text{ such that }x\geq 0,\ F(x,u)- v\geq 0, \text{ and }(F(x,u)- v)^T\cdot x=0,$$ where these are vector inequalities. We characterize the local upper Lipschitz continuity of the (possibly set-valued) solution mapping which assigns solutions $x$ to each parameter pair $(v,u)$. We also characterize when this solution mapping is locally a single-valued Lipschitzian mapping (so solutions exist, are unique, and depend Lipschitz continuously on the parameters). These characterizations are automatically sufficient conditions for the more general (and usual) case where $v= 0$. Finally, we study the differentiability properties of the solution mapping in both the single-valued and set-valued cases, in particular obtaining a new characterization of $B$-differentiability in the single-valued case, along with a formula for the $B$-derivative. Though these results cover a broad range of stability properties, they are all derived from similar fundamental principles of variational analysis.

MSC 2000:: *90C33 Complementarity problems
90C31 Sensitivity, etc.

Keywords: parameterized nonlinear complementarity problems; solution stability; $B$-derivatives; Lipschitz continuity; local upper Lipschitz continuity

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