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This paper is devoted to the study of the metric subregularity constraintqualification (MSCQ) for general optimization problems, with the emphasis onthe nonconvex setting. We elaborate on notions of directional pseudo- andquasi-normality, recently introduced by Bai et al. (SIAM J. Opt., 2019), whichcombine the standard approach via pseudo- and quasi-normality with modern toolsof directional variational analysis. We focus on applications to disjunctiveprograms, where (directional) pseudo-normality is characterized via an extremalcondition. This, in turn, yields efficient tools to verify pseudo-normality andMSCQ, which include, but are not limited to, Robinson's result on polyhedralmultifunctions and Gfrerer's second-order sufficient condition for metricsubregularity. Finally, we refine our study by defining the new class ofortho-disjunctive programs which comprises prominent optimization problems suchas mathematical programs with complementarity, vanishing or switchingconstraints.

Sufficient Conditions for Metric Subregularity of Constraint Systems with Applications to Disjunctive and Ortho-Disjunctive Programs