Mathematical Programming

, Volume 85, Issue 2, pp 397-406

First online:

Stability of solutions to parameterized nonlinear complementarity problems

  • A.B. LevyAffiliated withDepartment of Mathematics, Bowdoin College, Brunswick, ME 04011, USA

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such that x≥0,  F(x,u)-v≥0 , 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.

Key words: parameterized nonlinear complementarity problems – solution stability – B-derivatives – Lipschitz continuity – local upper Lipschitz continuity