Enhanced Fritz John Stationarity, New Constraint Qualifications and Local Error Bound for Mathematical Programs with Vanishing Constraints

TL;DR

This paper advances the understanding of mathematical programs with vanishing constraints by introducing weaker constraint qualifications, deriving new stationary conditions, and establishing local error bounds, thereby broadening the theoretical foundation for solving MPVC problems.

Contribution

It introduces a new weaker constraint qualification called MPVC-generalized quasinormality, and derives enhanced Fritz John stationary conditions and local error bounds for MPVC.

Findings

Enhanced Fritz John stationary conditions for MPVC.

Introduction of MPVC-generalized quasinormality as a weaker constraint qualification.

Establishment of local error bounds under the new qualification.

Abstract

In this paper, we study the difficult class of optimization problems called the mathematical programs with vanishing constraints or MPVC. Extensive research has been done for MPVC regarding stationary conditions and constraint qualifications using geometric approaches. We use the Fritz John approach for MPVC to derive the M-stationary conditions under weak constraint qualifications. An enhanced Fritz John type stationary condition is also derived for MPVC, which provides the notion of enhanced M-stationarity under a new and weaker constraint qualification: MPVC-generalized quasinormality. We show that this new constraint qualification is even weaker than MPVC-CPLD. A local error bound result is also established under MPVC-generalized quasinormality.