Exact penalty functions and global saddle points of augmented Lagrangians for well-posed constrained optimization problems

TL;DR

This paper establishes new verifiable conditions for the exactness of penalty functions and the existence of global saddle points of augmented Lagrangians in infinite dimensional constrained optimization problems, extending finite-dimensional results.

Contribution

It introduces a new version of extended well-posedness, enabling the extension of finite-dimensional results to infinite-dimensional spaces without restrictive assumptions.

Findings

Provides verifiable sufficient conditions for penalty function exactness

Establishes existence of global saddle points in infinite dimensions

Extends finite-dimensional results to infinite-dimensional settings

Abstract

The goal of this article is to study necessary and sufficient conditions for the exactness of penalty functions and the existence of global saddle points of augmented Lagrangians for well-posed (in a suitable sense) constrained optimization problems in infinite dimensional spaces. To this end, we present a new version of extended well-posedness of a constrained optimization problem and analyse how it relates to the more well-known types of well-posedness, such as Tykhonov and Levitin-Polyak well-posedness. This new version of extended well-posedness allows one to extend many existing results on exact penalty functions and global saddle points of augmented Lagrangians from the finite dimensional to the infinite dimensional case. Such extensions provide first verifiable sufficient conditions for the exactness of penalty functions and the existence of global saddle points of augmented Lagrangians in the infinite dimensional case that do not rely on very restrictive and difficult to verify assumptions (nonlocal metric regularity of constraints, existence of nonlocal error bounds, the Palais-Smale condition, abstract properties of the perturbation function, etc.) that are typically used in the literature.