  In this paper we consider a sufficiently broad class of nonlinear
mathematical programs with disjunctive constraints, which, e.g., include
mathematical programs with complemetarity/vanishing constraints. We present an
extension of the concept of ${\mathcal Q}$-stationarity as introduced in the
recent paper [2]. ${\mathcal Q}$-stationarity can be easily combined with the
well-known notion of M-stationarity to obtain the stronger property of
so-called ${\mathcal Q}_M$-stationarity. We show how the property of ${\mathcal
Q}_M$-stationarity (and thus also of M-stationarity) can be efficiently
verified for the considered problem class by computing ${\mathcal
Q}$-stationary solutions of a certain quadratic program. We consider further
the situation that the point which is to be tested for ${\mathcal
Q}_M$-stationarity, is not known exactly, but is approximated by some
convergent sequence, as it is usually the case when applying some numerical
method.
