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Stability of solutions to parameterized nonlinear complementarity problems. (English) Zbl 0955.90138

Summary: We consider the stability properties of solutions to parameterized nonlinear complementarity problems

Findx n suchthatx0,F(x,u)-v0,and(F(x,u)-v) T ·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.

90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
90C31Sensitivity, stability, parametric optimization