Existence of augmented Lagrange multipliers: reduction to exact penalty functions and localization principle

TL;DR

This paper establishes a theoretical link between augmented Lagrange multipliers and exact penalty functions, introducing a localization principle to analyze their existence through local behavior near optimal solutions.

Contribution

It provides new general results on the existence of augmented Lagrange multipliers, connecting them with penalty function exactness and introducing the localization principle for their study.

Findings

Existence of augmented Lagrange multipliers is equivalent to penalty function exactness under certain conditions.

The localization principle enables analysis of augmented Lagrangian behavior near optimal solutions.

New theoretical framework for studying augmented Lagrange multipliers in constrained optimization.

Abstract

In this article, we present new general results on existence of augmented Lagrange multipliers. We define a penalty function associated with an augmented Lagrangian, and prove that, under a certain growth assumption on the augmenting function, an augmented Lagrange multiplier exists if and only if this penalty function is exact. We also develop a new general approach to the study of augmented Lagrange multipliers called the localization principle. The localization principle allows one to study the local behaviour of the augmented Lagrangian near globally optimal solutions of the initial optimization problem in order to prove the existence of augmented Lagrange multipliers.