# Metric Regularity of Quasidifferentiable Mappings and Optimality Conditions for Nonsmooth Mathematical Programming Problems

## TL;DR

This paper develops new conditions for metric regularity of quasidifferentiable mappings and applies them to improve optimality conditions in nonsmooth mathematical programming problems.

## Contribution

It introduces novel necessary and sufficient conditions for metric regularity using quasidifferentials and proposes a new constraint qualification for quasidifferentiable systems.

## Key findings

- New conditions for local metric regularity in terms of quasidifferentials.
- A new MFCQ-type constraint qualification ensuring metric regularity.
- Enhanced optimality conditions that can detect non-optimal points where traditional conditions fail.

## Abstract

This article is devoted to the analysis of necessary and/or sufficient conditions for metric regularity in terms of Demyanov-Rubinov-Polyakova quasidifferentials. We obtain new necessary and sufficient conditions for the local metric regularity of a multifunction in terms of quasidifferentials of the distance function to this multifunction. We also propose a new MFCQ-type constraint qualification for a parametric system of quasidifferentiable equality and inequality constraints and prove that it ensures the metric regularity of a multifunction associated with this system. As an application, we utilize our constraint qualification to strengthen existing optimality conditions for quasidifferentiable programming problems with equality and inequality constraints. We also prove the independence of the optimality conditions of the choice of quasidifferentials and present a simple example in which the optimality conditions in terms of quasidifferentials detect the non-optimality of a given point, while optimality conditions in terms of various subdifferentials fail to disqualify this point as non-optimal.