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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

$\text{Find}\phantom{\rule{4.pt}{0ex}}x\in {ℝ}^{n}\phantom{\rule{4.pt}{0ex}}\text{such}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{4.pt}{0ex}}x\ge 0,\phantom{\rule{4pt}{0ex}}F\left(x,u\right)-v\ge 0,\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{\left(F\left(x,u\right)-v\right)}^{T}·x=0,$

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 $\left(v,u\right)$. 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:
 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions) 90C31 Sensitivity, stability, parametric optimization