A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

Abstract

In this two-part study we develop a unified approach to the analysis of the global exactness of various penalty and augmented Lagrangian functions for constrained optimization problems in finite dimensional spaces. This approach allows one to verify in a simple and straightforward manner whether a given penalty/augmented Lagrangian function is exact, i.e. whether the problem of unconstrained minimization of this function is equivalent (in some sense) to the original constrained problem, provided the penalty parameter is sufficiently large. Our approach is based on the so-called localization principle that reduces the study of global exactness to a local analysis of a chosen merit function near globally optimal solutions. In turn, such local analysis can usually be performed with the use of sufficient optimality conditions and constraint qualifications.In the first paper we introduce the concept of global parametric exactness and derive the localization principle in the parametric form. With the use of this version of the localization principle we recover existing simple necessary and sufficient conditions for the global exactness of linear penalty functions, and for the existence of augmented Lagrange multipliers of Rockafellar-Wets' augmented Lagrangian. Also, we obtain completely new necessary and sufficient conditions for the global exactness of general nonlinear penalty functions, and for the global exactness of a continuously differentiable penalty function for nonlinear second-order cone programming problems. We briefly discuss how one can construct a continuously differentiable exact penalty function for nonlinear semidefinite programming problems, as well.

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

“…This principle allows one to reduce the study of the global exactness of the separating function F (x, λ, c) to a local analysis of behaviour of this function near globally optimal solutions of the problem (P). Below, we follow the same line of reasoning as during the derivation of the localization principle in the parametric form in the first part of our study (see [15]).…”

“…Furthermore, taking into accout the facts that DG(x * )w * ∈ C 0 (x * , λ * ), ∇h(x * )w * = 0 and ∇ x L(ξ * ) = 0 (recall that ξ * is a KKT-point) one obtains that DG(x * )w * ∈ T K (G(x * )) and ∇f (x * ), w * = 0. Hence with the use of the description of the contingent cone T K (G(x * )) in terms of the mapping G(x * ) [2, Example 2.65] one gets that w * belongs to the critical cone (15), which contradicts the fact that the second order sufficient optimality condition holds true at x * . Thus, the augmented Lagrangian function L (ξ, c) is locally exact at ξ * .…”

“…A motivation behind the study of the exactness of penalty/augmented Lagrangian functions, a review of the relevant literature, as well as some general discussions of the framework for the study of global exactness that is adopted in our research, are presented in the first paper of the series (see the preprint [15]).…”

Minimax exactness and global saddle points of nonlinear augmented Lagrangians

“…As was shown in [16,24], one can formulate a general principle, called the localization principle, that allows one to obtain all known results on the existence of global saddle points/augmented Lagrange multipliers in the finite dimensional case in a unified manner. However, the results of [16,24] cannot be applied to nonlinear (in the objective function) augmented Lagrangians that were studied in, e.g., [9,23,27] In [28,29], a unified theory of various concepts of exactness of penalty and augmented Lagrangian functions was developed. This theory provides a unified approach to an analysis of linear [30,31,32,33] and nonlinear [34,35,36] exact penalty functions, Fletcher's continuously differentiable exact penalty functions [37,38,39,40], exact augmented Lagrangians [16,41,42,43], Huyer and Neumaier's exact penalty functions [44,45], and the existence of augmented Lagrange multipliers in the finite dimensional case.…”

A new logarithmic penalty function approach for nonlinear constrained optimization problem

“…Other researchers (see for instance (Echebest et al, 2016;Dolgopolik, 2018) further investigated exponential penalty function in connection with augmented Lagrangian functions. Nevertheless, most of the existing penalty functions are mainly applicable to inequality constraints only.…”