A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions II: Extended Exactness

TL;DR

This paper introduces the concept of global extended exactness for penalty and augmented Lagrangian functions, providing a localization principle that simplifies the analysis of their global properties in constrained optimization.

Contribution

It extends the theory of penalty and augmented Lagrangian functions by incorporating extra variables, enabling the design of smooth, globally exact functions for complex problems.

Findings

Derived necessary and sufficient conditions for global extended exactness.

Applied the localization principle to nonlinear semidefinite programming.

Constructed a smooth, globally exact augmented Lagrangian for specific problems.

Abstract

In the second part of our study we introduce the concept of global extended exactness of penalty and augmented Lagrangian functions, and derive the localization principle in the extended form. The main idea behind the extended exactness consists in an extension of the original constrained optimization problem by adding some extra variables, and then construction of a penalty/augmented Lagrangian function for the extended problem. This approach allows one to design extended penalty/augmented Lagrangian functions having some useful properties (such as smoothness), which their counterparts for the original problem might not possess. In turn, the global exactness of such extended merit functions can be easily proved with the use of the localization principle presented in this paper, which reduces the study of global exactness to a local analysis of a merit function based on sufficient optimality conditions and constraint qualifications. We utilize the localization principle in order to obtain simple necessary and sufficient conditions for the global exactness of the extended penalty function introduced by Huyer and Neumaier, and in order to construct a globally exact continuously differentiable augmented Lagrangian function for nonlinear semidefinite programming problems.