Criticality of Lagrange Multipliers in Variational Systems

TL;DR

This paper investigates the criticality of Lagrange multipliers in general nonpolyhedral variational systems, especially those related to conic programming, using advanced second-order variational analysis techniques.

Contribution

It introduces a novel approach based on second-order variational analysis to characterize noncritical multipliers in nonpolyhedral systems, extending beyond polyhedral cases.

Findings

Complete characterizations of noncritical multipliers in nonpolyhedral systems.

Application of results to semidefinite programming examples.

Overcoming challenges of nonpolyhedrality with advanced analysis techniques.

Abstract

The paper concerns the study of criticality of Lagrange multipliers in variational systems that has been recognized in both theoretical and numerical aspects of optimization and variational analysis. In contrast to the previous developments dealing with polyhedral KKT systems and the like, we now focus on general nonpolyhedral systems that are associated, in particular, with problems of conic programming. Developing a novel approach, which is mainly based on advanced techniques and tools of second-order variational analysis and generalized differentiation, allows us to overcome principal challenges of nonpolyhedrality and to establish complete characterizations on noncritical multipliers in such settings. The obtained results are illustrated by examples from semidefinite programming.