Variational Conditions Under the Constant Rank Constraint Qualification

This paper studies solution properties of a parametric variational condition under the constant rank constraint qualification (CRCQ), and properties of its underlying set. We start by showing that if the CRCQ holds at a point in a fixed set, then there exists a one-to-one correspondence between the collection of nonempty faces of the normal cone to the set at that point and the collection of active index sets at points nearby. We then study the behavior of the Euclidean projector, and prove under the CRCQ that the set of multipliers associated with the Euclidean projection is locally a continuous multifunction. Following that, we apply the degree theory to a localized normal map, to show that the combination of the CRCQ and the so-called strong coherent orientation condition suffices for the parametric variational condition to have a locally unique, continuous solution, which is selected from finitely many C¹ functions. Applications of this result under additional assumptions extend or recover some earlier results on parametric variational inequalities.