Augmented Lagrangian Functions for Cone Constrained Optimization: the Existence of Global Saddle Points and Exact Penalty Property

TL;DR

This paper develops a unified theory for augmented Lagrangian functions in cone constrained optimization, introducing a localization principle for saddle point existence and constructing globally exact functions to improve optimization methods.

Contribution

It introduces a new localization principle for proving the existence of saddle points and constructs globally exact augmented Lagrangians for various cone constrained problems.

Findings

Unified framework for augmented Lagrangians in cone constrained problems.

Localization principle reduces saddle point existence to local optimality analysis.

Constructed globally exact augmented Lagrangians for multiple problem classes.

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.