Stability of solutions to parameterized nonlinear complementarity problems

We consider the stability properties of solutions to parameterized nonlinea r complementarity problems Find x is an element of R-n such that x greater than or equal to 0, F(x, u) - v greater than or equal to 0, and (F(x, u) - v)(T) . x = 0 where these are vector inequalities. We characterize the local upper Lipsch itz continuity of the (possibly set-valued) solution mapping which assigns solutions x to each parameter pair (v, u). We also characterize when this s olution mapping is locally a single-valued Lipschitzian mapping (so solutio ns exist, are unique, and depend Lipschitz continuously on the parameters). These characterizations are automatically sufficient conditions for the mo re general land usual) case where v = 0. Finally, we study the differentiab ility properties of the solution mapping in both the single-valued and set- valued cases, in particular obtaining a new characterization of B-different iability in the single-valued case, along with a formula for the B-derivati ve. Though these results cover a broad range of stability properties, they are all derived from similar fundamental principles of variational analysis .