  When dealing with general Lipschitzian optimization problems, there are many
problem classes where standard constraint qualifications fail at local
minimizers. In contrast to a constraint qualification, a problem qualification
does not only rely on the constraints but also on the objective function to
guarantee that a local minimizer is a Karush-Kuhn-Tucker (KKT) point. For
example, calmness in the sense of Clarke is a problem qualification. In this
article, we introduce the Subset Mangasarian-Fromovitz Condition (subMFC). This
new problem qualification is based on a nonsmooth version of the approximate
KKT conditions, which hold at every local minimizer without further
assumptions. A comparison with existing constraint qualifications and problem
qualifications for the given problem class reveals that subMFC is strictly
weaker than quasinormality and can hold even if the local error bound
condition, the cone-continuity property, Guignard's constraint qualification
and calmness in the sense of Clarke are violated. Further, we emphasize the
power of the new problem qualification within the context of bilevel
optimization. More precisely, under mild assumptions on the problem data, we
suggest a version of subMFC that is tailored to the lower-level value function
reformulation. It turns out that this new condition can be satisfied even if
the widely used partial calmness condition does not hold.
