Relations between Abs-Normal NLPs and MPCCs. Part 2: Weak Constraint Qualifications

TL;DR

This paper explores the relationship between abs-normal nonlinear programming problems and MPCCs, focusing on weak constraint qualifications, and introduces new stationarity concepts and optimality conditions.

Contribution

It establishes the preservation of certain constraint qualifications and introduces new stationarity concepts for abs-normal NLPs in relation to MPCC reformulations.

Findings

Constraint qualifications of Abadie type are preserved.

Guginard's and Abadie's constraint qualifications are equivalent for all branch problems.

Introduces M-stationarity and B-stationarity concepts with first-order optimality conditions.

Abstract

This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their relation to equivalent MPCC reformulations. We introduce the concepts of Abadie's and Guignard's kink qualification and prove relations to MPCC-ACQ and MPCC-GCQ for the counterpart MPCC formulations. Due to non-uniqueness of a specific slack reformulation suggested in [10], the relations are non-trivial. It turns out that constraint qualifications of Abadie type are preserved. We also prove the weaker result that equivalence of Guginard's (and Abadie's) constraint qualifications for all branch problems hold, while the question of GCQ preservation remains open. Finally, we introduce M-stationarity and B-stationarity concepts for abs-normal NLPs and prove first order optimality conditions corresponding to MPCC counterpart formulations.