  In this paper we study sufficient conditions for metric subregularity of a
set-valued map which is the sum of a single-valued continuous map and a locally
closed subset. First we derive a sufficient condition for metric subregularity
which is weaker than the so-called first-order sufficient condition for metric
subregularity (FOSCMS) by adding an extra sequential condition. Then we
introduce a directional version of the quasi-normality and the pseudo-normality
which is stronger than the new {weak} sufficient condition for metric
subregularity but is weaker than the classical quasi-normality and
pseudo-normality respectively. Moreover we introduce a nonsmooth version of the
second-order sufficient condition for metric subregularity and show that it is
a sufficient condition for the new sufficient condition for metric
{sub}regularity to hold. An example is used to illustrate that the directional
pseduo-normality can be weaker than FOSCMS. For the class of set-valued maps
where the single-valued mapping is affine and the abstract set is the union of
finitely many convex polyhedral sets, we show that the pseudo-normality and
hence the directional pseudo-normality holds automatically at each point of the
graph. Finally we apply our results to the complementarity and the
Karush-Kuhn-Tucker systems.
