Relation between the constant rank and the relaxed constant rank constraint qualifications

This article discusses a relation between the constant rank constraint qualification (CRCQ) and the recently proposed relaxed constant rank constraint qualification (RCRCQ). We show that a parametric constraint system satisfying the RCRCQ is locally diffeomorphic to a system satisfying the CRCQ. We use this result to extend some existing results for the CRCQ to the RCRCQ, establish a relation between the RCRCQ and the Mangasarian–Fromovitz constraint qualification, and obtain a weakened version of the Aubin property under the RCRCQ.