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

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.