Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property

Abstract

In this 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 these problems 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 the existence of global saddle points, and, in essence, reduces the problem of the existence of global 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.

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

“…The first exact augmented Lagrangian function was introduced by Di Pillo and Grippo in [3], and later on was improved and thoroughly investigated by many researchers [3,5,4,27,10,8,9,7,6,19,18,28,11,20]. The general theory of exact augmented Lagrangian functions for cone constrained optimization problems was developed by the author in [17].…”

“…The main goal of this section is to introduce a continuously differentiable exact augmented Lagrangian function for nonlinear semidefinite programming problem, and to prove its global extended exactness with the use of the localization principle. This augmented Lagrangian function was first introduced by the author in [17]; however, the paper [17] does not contain a proof of the global exactness of this augmented Lagrangian. Here we present a detailed and almost self-contained (apart from some technical results from semidefinite optimization) proof of this result.…”

“…Recently, Di Pillo and Grippo's exact augmented Lagrangian function was extended to the case of nonlinear semidefinite programming problems [20]. A general theory of globally exact augmented Lagrangian functions for cone constrained optimization problems was developed by the author in [17]. It should be noted that the main feature of exact augmented Lagrangian functions is the fact that one has to minimize these function in primal and dual variables simultaneously in order to compute KKT-points corresponding to globally optimal solution of the original constrained problem.…”

“…With the use of this principle we recover existing simple necessary and sufficient conditions for the global exactness of Huyer and Neumaier's extended penalty function. We also study the global exactness of a continuously differentiable augmented Lagrangian func-tion for nonlinear semidefinite programming problems that was recently introduced by the author [17]. It should be noted that the theorem on the global exactness of this augmented Lagrangian function was formulated in [17] without proof.…”

“…We also study the global exactness of a continuously differentiable augmented Lagrangian func-tion for nonlinear semidefinite programming problems that was recently introduced by the author [17]. It should be noted that the theorem on the global exactness of this augmented Lagrangian function was formulated in [17] without proof. In this paper we present a detailed proof of this result.…”