  This paper is devoted to the study of the metric subregularity constraint
qualification (MSCQ) for general optimization problems, with the emphasis on
the nonconvex setting. We elaborate on notions of directional pseudo- and
quasi-normality, recently introduced by Bai et al. (SIAM J. Opt., 2019), which
combine the standard approach via pseudo- and quasi-normality with modern tools
of directional variational analysis. We focus on applications to disjunctive
programs, where (directional) pseudo-normality is characterized via an extremal
condition. This, in turn, yields efficient tools to verify pseudo-normality and
MSCQ, which include, but are not limited to, Robinson's result on polyhedral
multifunctions and Gfrerer's second-order sufficient condition for metric
subregularity. Finally, we refine our study by defining the new class of
ortho-disjunctive programs which comprises prominent optimization problems such
as mathematical programs with complementarity, vanishing or switching
constraints.
