# Criticality of Lagrange Multipliers in Extended Nonlinear Optimization

## TL;DR

This paper investigates the criticality of Lagrange multipliers in extended nonlinear programming, providing a systematic variational analysis framework that links multiplier criticality with stability concepts using advanced second-order tools.

## Contribution

It introduces the first comprehensive study of multiplier criticality in ENLP, offering verifiable characterizations and relationships with stability notions in variational analysis.

## Key findings

- Characterization of critical and noncritical multipliers in ENLP
- Relationships between multiplier criticality and stability concepts
- Application of second-order variational analysis tools

## Abstract

The paper is devoted to the study and applications of criticality of Lagrange multipliers in variational systems, which are associated with the class of problems in composite optimization known as extended nonlinear programming (ENLP). The importance of both ENLP and the concept of multiplier criticality in variational systems has been recognized in theoretical and numerical aspects of optimization and variational analysis, while the criticality notion has never been investigated in the ENLP framework. We present here a systematic study of critical and noncritical multipliers in a general variational setting that covers, in particular, KKT systems in ENLP with establishing their verifiable characterizations as well as relationships between noncriticality and other stability notions in variational analysis. Our approach is mainly based on advanced tools of second-order variational analysis and generalized differentiation.