Characterizations of strong regularity for variational inequalities over polyhedral convex sets.

*(English)*Zbl 0899.49004Linear and nonlinear variational inequality problems over a polyhedral set \(C\) are considered. When \(C\) is the nonnegative orthant the inequalities correspond to nonlinear and linear complementarity problems. The problems are analyzed parametrically. Let \(L(q)\) denote the set of solutions to a linearized variational inequality (at a given point) in dependence on an additive parameter vector \(q\). The strong regularity property of a variational inequality is provided by local single-valuedness and Lipschitz continuity of \(L(q)\). The property garantees the same Lipschitz stability for a non-linearized inequality. In the paper, the strong regularity is characterized through the pseudo-Lipschitz continuity by Aubin and lower semicontinuity of \(L\). In the linear case, the lower semicontinuity of \(L\) everywhere is equivalent to the single-valuedness of \(L\) everywhere. The characterization of the strong regularity in the linear case is given through the critical face condition. Some applications to the complementarity problem and nonlinear programming are given. In particular, the critical face condition gives a new characterization for local Lipschitz continuity and the single valuedness of the map defined by Karush-Kuhn-Tucker conditions.

Reviewer: N.Novikova (Moskva)

##### MSC:

49J40 | Variational inequalities |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

49K40 | Sensitivity, stability, well-posedness |

90C31 | Sensitivity, stability, parametric optimization |

90C25 | Convex programming |