Asymptotic stationarity and regularity for nonsmooth optimization problems

TL;DR

This paper develops a new optimality condition for nonsmooth optimization problems using variational analysis, introducing AM-regularity as a key constraint qualification to ensure stationarity aligns with Mordukhovich-stationarity.

Contribution

It introduces AM-regularity, a novel constraint qualification, and explores its relationship with existing qualifications, enhancing understanding of stationarity in nonsmooth optimization.

Findings

AM-regularity ensures stationarity aligns with Mordukhovich-stationarity.

AM-regularity recovers cone-continuity-type constraint qualifications.

Application to geometric and disjunctive constraints demonstrates practical relevance.

Abstract

Based on the tools of limiting variational analysis, we derive a sequential necessary optimality condition for nonsmooth mathematical programs which holds without any additional assumptions. In order to ensure that stationary points in this new sense are already Mordukhovich-stationary, the presence of a constraint qualification which we call AM-regularity is necessary. We investigate the relationship between AM-regularity and other constraint qualifications from nonsmooth optimization like metric (sub-)regularity of the underlying feasibility mapping. Our findings are applied to optimization problems with geometric and, particularly, disjunctive constraints. This way, it is shown that AM-regularity recovers recently introduced cone-continuity-type constraint qualifications, sometimes referred to as AKKT-regularity, from standard nonlinear and complementarity-constrained optimization. Finally, we discuss some consequences of AM-regularity for the limiting variational calculus.