Second-Order Necessary Conditions, Constraint Qualifications and Exact Penalty for Mathematical Programs with Switching Constraints

TL;DR

This paper develops second-order necessary conditions, constraint qualifications, and exact penalty results for mathematical programs with switching constraints, advancing theoretical understanding and solution characterization.

Contribution

It introduces new second-order constraint qualifications and characterizations for stationary points in MPSC, and establishes local exact penalty results under mild conditions.

Findings

New second-order constraint qualifications for MPSC

Characterizations of Mordukhovich and strong stationary points

Conditions for local exact penalty in MPSC

Abstract

In this paper, we investigate second-order necessary conditions and exact penalty of mathematical programs with switching constraints (MPSC). Some new second-order constraint qualifications and second-order quasi-normality are introduced for (MPSC), which are crucial to establish the second-order necessary conditions and the error bound of (MPSC). We explore the relations among these constraint qualifications in term of (MPSC). The characterizations of Mordukhovich stationary point and strong stationary point of (MPSC) are derived under some mild conditions. A sufficient condition is provided for a Mordukhovich stationary point of (MPSC) being a strong stationary point. The strong second-order necessary conditions as well as weak second-order necessary conditions of (MPSC) are established under these weak constraint qualifications. Finally, we obtain the local exact penalty of (MPSC) under the local error bound or some constraint qualifications in term of (MPSC).