Relations between Abs-Normal NLPs and MPCCs. Part 1: Strong Constraint Qualifications

TL;DR

This paper explores the relationship between abs-normal NLPs and MPCCs, demonstrating the equivalence of their constraint qualifications and stationarity conditions, thus advancing the understanding of non-smooth optimization problem reformulations.

Contribution

It establishes the equivalence of constraint qualifications and optimality conditions between abs-normal NLPs and MPCCs, including analysis of slack reformulations.

Findings

Kink qualifications and MPCC constraint qualifications are equivalent.

Strong stationarity concepts are equivalent for both problem classes.

Slack reformulations preserve some constraint qualifications but not others.

Abstract

This work is part of an ongoing effort of comparing non-smooth optimization problems in abs-normal form to MPCCs. We study the general abs-normal NLP with equality and inequality constraints in relation to an equivalent MPCC reformulation. We show that kink qualifications and MPCC constraint qualifications of linear independence type and Mangasarian-Fromovitz type are equivalent. Then we consider strong stationarity concepts with first and second order optimality conditions, which again turn out to be equivalent for the two problem classes. Throughout we also consider specific slack reformulations suggested in [9], which preserve constraint qualifications of linear independence type but not of Mangasarian-Fromovitz type.