New stationarity conditions between strong and M-stationarity for mathematical programs with complementarity constraints

TL;DR

This paper introduces new first-order necessary conditions for MPCCs that are between strong and M-stationarity, applicable under weak constraint qualifications, and extends these results to MPVCs with simpler proofs.

Contribution

It proposes new stationarity conditions for MPCCs that are simpler and lie between existing concepts, valid under weak constraint qualifications, and extends the approach to MPVCs.

Findings

New stationarity conditions between strong and M-stationarity for MPCCs.

Conditions hold for local minimizers under MPCC-GCQ.

Simplified proof of M-stationarity for MPVCs.

Abstract

We introduce new first-order necessary conditions for mathematical programs with complementarity constraints (MPCCs), which lie between strong and M-stationarity and have a relatively simple description. We show that they hold for local minimizers under the rather weak constraint qualification MPCC-GCQ. As a generalization, we also get a class of stationarity conditions that lie between strong and C-stationarity and show that they also hold for local minimizers under MPCC-GCQ. We also present similar results for mathematical programs with vanishing constraints (MPVCs), and a very simple and elementary proof of M-stationarity for local minimizers of MPVCs.