On the directional asymptotic approach in optimization theory Part B: constraint qualifications

TL;DR

This paper advances the theory of asymptotic constraint qualifications in nonsmooth optimization by incorporating directional data, introducing new regularity conditions, and providing tools for their analysis.

Contribution

It introduces directional asymptotic regularity conditions and a coderivative-like tool, extending the understanding of constraint qualifications in nonsmooth optimization.

Findings

Directional asymptotic regularity conditions are comparable to standard qualifications.

New concepts of pseudo- and quasi-normality are introduced for set-valued mappings.

The proposed tools can be computed from initial problem data for geometric constraints.

Abstract

During the last years, asymptotic (or sequential) constraint qualifications, which postulate upper semicontinuity of certain set-valued mappings and provide a natural companion of asymptotic stationarity conditions, have been shown to be comparatively mild, on the one hand, while possessing inherent practical relevance from the viewpoint of numerical solution methods, on the other one. Based on recent developments, the theory in this paper enriches asymptotic constraint qualifications for very general nonsmooth optimization problems over inverse images of set-valued mappings by incorporating directional data. We compare these new directional asymptotic regularity conditions with standard constraint qualifications from nonsmooth optimization. Further, we introduce directional concepts of pseudo- and quasi-normality which apply to set-valued mappings. It is shown that these properties provide sufficient conditions for the validity of directional asymptotic regularity. Finally, a novel coderivative-like variational tool is introduced which allows to study the presence of directional asymptotic regularity. For geometric constraints, it is illustrated that all appearing objects can be calculated in terms of initial problem data.