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 finite-dimensional constrained optimization problems. 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.

34 pages. arXiv admin note: text overlap with arXiv:1710.01961

References in corpus (8)

• A Unifying Theory of Exactness of Linear Penalty Functions
• A unifying theory of exactness of linear penalty functions II: parametric penalty functions
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• Smooth exact penalty functions II: a reduction to standard exact penalty functions
• Augmented Lagrangian Functions for Cone Constrained Optimization: the Existence of Global Saddle Points and Exact Penalty Property
• A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness
• A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions II: Extended Exactness

Cited by in corpus (4)

• Exact penalty functions with multidimensional penalty parameter and adaptive penalty updates
• A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness
• New global optimality conditions for nonsmooth DC optimization problems
• A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions II: Extended Exactness