Parametric Variational Inequalities with Multivalued Solution Sets

In this paper we study basic properties of multivalued solution sets of parametric variational inequalities under the assumption that the unperturbed problem has a locally unique solution. We derive general existence and Lipschitz continuity results, which extend previous developments, for multivalued solution set maps of parametric variational inequalities defined on polyhedral sets. These results, when applied to parametric nonlinear complementarity problems, yield existence and Lipschitz continuity of multivalued solutions under new conditions on the Jacobian matrix of the variational function. We also obtain a novel result on the existence of (nonunique) solutions to parametric variational inequalities defined on perturbed sets. Finally, we show that the recent directional differentiability results of Qiu and Magnanti can be derived in a unified manner under somewhat more general assumptions which, in particular, makes them applicable to nonlinear complementarity problems.