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

## TL;DR

This paper investigates metric subregularity constraint qualification for nonconvex optimization problems, introducing directional pseudo- and quasi-normality notions, and applies these to disjunctive and ortho-disjunctive programs to develop verification tools.

## Contribution

It introduces new directional normality concepts and applies them to disjunctive and ortho-disjunctive programs, enhancing the understanding and verification of metric subregularity.

## Key findings

- Characterization of pseudo-normality via extremal conditions
- Development of tools to verify MSCQ in complex programs
- Extension of existing conditions to new classes like ortho-disjunctive programs

## Abstract

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.