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In the second part of our study we introduce the concept of global extendedexactness of penalty and augmented Lagrangian functions, and derive thelocalization principle in the extended form. The main idea behind the extendedexactness consists in an extension of the original constrained optimizationproblem by adding some extra variables, and then construction of apenalty/augmented Lagrangian function for the extended problem. This approachallows one to design extended penalty/augmented Lagrangian functions havingsome useful properties (such as smoothness), which their counterparts for theoriginal problem might not possess. In turn, the global exactness of suchextended merit functions can be easily proved with the use of the localizationprinciple presented in this paper, which reduces the study of global exactnessto a local analysis of a merit function based on sufficient optimalityconditions and constraint qualifications. We utilize the localization principlein order to obtain simple necessary and sufficient conditions for the globalexactness of the extended penalty function introduced by Huyer and Neumaier,and in order to construct a globally exact continuously differentiableaugmented Lagrangian function for nonlinear semidefinite programming problems.

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