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In this paper we consider a sufficiently broad class of nonlinearmathematical programs with disjunctive constraints, which, e.g., includemathematical programs with complemetarity/vanishing constraints. We present anextension of the concept of ${\mathcal Q}$-stationarity as introduced in therecent paper [2]. ${\mathcal Q}$-stationarity can be easily combined with thewell-known notion of M-stationarity to obtain the stronger property ofso-called ${\mathcal Q}_M$-stationarity. We show how the property of ${\mathcalQ}_M$-stationarity (and thus also of M-stationarity) can be efficientlyverified for the considered problem class by computing ${\mathcalQ}$-stationary solutions of a certain quadratic program. We consider furtherthe situation that the point which is to be tested for ${\mathcalQ}_M$-stationarity, is not known exactly, but is approximated by someconvergent sequence, as it is usually the case when applying some numericalmethod.

New verifiable stationarity concepts for a class of mathematical programs with disjunctive constraints