New verifiable stationarity concepts for a class of mathematical programs with disjunctive constraints

TL;DR

This paper introduces new verifiable stationarity concepts for nonlinear mathematical programs with disjunctive constraints, enhancing the analysis and verification of solutions in complex optimization problems.

Contribution

It extends the ${\mathcal Q}$-stationarity concept to ${\mathcal Q}_M$-stationarity, providing practical methods for verification and handling approximate solutions.

Findings

${\mathcal Q}_M$-stationarity can be efficiently verified.

The approach applies to problems with complementarity and vanishing constraints.

Verification remains robust under solution approximation.

Abstract

In this paper we consider a sufficiently broad class of nonlinear mathematical programs with disjunctive constraints, which, e.g., include mathematical programs with complemetarity/vanishing constraints. We present an extension of the concept of ${\mathcal Q}$-stationarity as introduced in the recent paper [2]. ${\mathcal Q}$-stationarity can be easily combined with the well-known notion of M-stationarity to obtain the stronger property of so-called ${\mathcal Q}_M$-stationarity. We show how the property of ${\mathcal Q}_M$-stationarity (and thus also of M-stationarity) can be efficiently verified for the considered problem class by computing ${\mathcal Q}$-stationary solutions of a certain quadratic program. We consider further the situation that the point which is to be tested for ${\mathcal Q}_M$-stationarity, is not known exactly, but is approximated by some convergent sequence, as it is usually the case when applying some numerical method.