A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness
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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 be performed with the use of 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. We also present 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.
KeywordsPenalty function Augmented Lagrangian function Exactness Localization principle Semidefinite programming
Mathematics Subject Classification65K05 90C30
The author is grateful to Professor Franco Giannessi for pointing out to the author the importance of development of a general theory of the exactness of penalty and augmented Lagrangian functions several years ago. The author also wishes to express his thanks to Professor X. Yang for thoughtful and stimulating comments that helped to improve the quality of the article.
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