  This article is devoted to the analysis of necessary and/or sufficient
conditions for metric regularity in terms of Demyanov-Rubinov-Polyakova
quasidifferentials. We obtain new necessary and sufficient conditions for the
local metric regularity of a multifunction in terms of quasidifferentials of
the distance function to this multifunction. We also propose a new MFCQ-type
constraint qualification for a parametric system of quasidifferentiable
equality and inequality constraints and prove that it ensures the metric
regularity of a multifunction associated with this system. As an application,
we utilize our constraint qualification to strengthen existing optimality
conditions for quasidifferentiable programming problems with equality and
inequality constraints. We also prove the independence of the optimality
conditions of the choice of quasidifferentials and present a simple example in
which the optimality conditions in terms of quasidifferentials detect the
non-optimality of a given point, while optimality conditions in terms of
various subdifferentials fail to disqualify this point as non-optimal.
