New Constraint Qualifications for Optimization Problems in Banach Spaces based on Asymptotic KKT Conditions

TL;DR

This paper introduces new constraint qualifications for optimization problems in Banach spaces based on asymptotic KKT conditions, addressing the scarcity and limitations of existing qualifications in infinite-dimensional settings.

Contribution

It extends the concept of asymptotic KKT regularity to Banach spaces, providing new constraint qualifications and analyzing their relation to existing ones.

Findings

New constraint qualifications are derived for Banach space optimization.

The paper demonstrates the applicability of these qualifications through examples.

An algorithmic application to augmented Lagrangian methods is presented.

Abstract

Optimization theory in Banach spaces suffers from the lack of available constraint qualifications. Despite the fact that there exist only a very few constraint qualifications, they are, in addition, often violated even in simple applications. This is very much in contrast to finite-dimensional nonlinear programs, where a large number of constraint qualifications is known. Since these constraint qualifications are usually defined using the set of active inequality constraints, it is difficult to extend them to the infinite-dimensional setting. One exception is a recently introduced sequential constraint qualification based on asymptotic KKT conditions. This paper shows that this so-called asymptotic KKT regularity allows suitable extensions to the Banach space setting in order to obtain new constraint qualifications. The relation of these new constraint qualifications to existing ones is discussed in detail. Their usefulness is also shown by several examples as well as an algorithmic application to the class of augmented Lagrangian methods.