Directional Quasi-/Pseudo-Normality as Sufficient Conditions for Metric Subregularity

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 directional versions of quasi-normality and pseudo-normality which are stronger than the new weak sufficient condition for metric subregularity but weaker than classical quasi-normality and pseudo-normality. 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 subregularity to hold. An example is used to illustrate that directional pseudo-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 pseudo-normality and hence directional pseudo-normality holds automatically at each point of the graph. Finally we apply our results to complementarity and Karush-Kuhn-Tucker systems.