Title: Augmented Lagrangian Functions for Cone Constrained Optimization: the Existence of Global Saddle Points and Exact Penalty Property,
Abstract: In the article we present a general theory of augmented Lagrangian functions
for cone constrained optimization problems that allows one to study almost all
known augmented Lagrangians for cone constrained programs within a unified
framework. We develop a new general method for proving the existence of global
saddle points of augmented Lagrangian functions, called the localization
principle. The localization principle unifies, generalizes and sharpens most of
the known results on existence of global saddle points, and, in essence,
reduces the problem of the existence of saddle points to a local analysis of
optimality conditions. With the use of the localization principle we obtain
first necessary and sufficient conditions for the existence of a global saddle
point of an augmented Lagrangian for cone constrained minimax problems via both
second and first order optimality conditions. In the second part of the paper,
we present a general approach to the construction of globally exact augmented
Lagrangian functions. The general approach developed in this paper allowed us
not only to sharpen most of the existing results on globally exact augmented
Lagrangians, but also to construct first globally exact augmented Lagrangian
functions for equality constrained optimization problems, for nonlinear second
order cone programs and for nonlinear semidefinite programs. These globally
exact augmented Lagrangians can be utilized in order to design new
superlinearly (or even quadratically) convergent optimization methods for cone
constrained optimization problems.