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

M. V. Dolgopolik

Published 2017-10-05Version 1

In the second paper of the series we introduce the concept of the global extended exactness of penalty and augmented Lagrangian functions, and derive the localization principle in the extended form. The main idea behind the extended exactness consists in the extension of the original constrained optimization problem by adding some extra variables, and then the construction of a penalty/augmented Lagrangian function for the extended problem. This approach allows one to design extended penalty/augmented Lagrangian functions having some useful properties (such as smoothness), which their counterparts for the original problem might not possess. In turn, the global exactness of such extended functions can be easily proved with the use of the localization principle presented in this paper, which reduces the study of global exactness to a local analysis of sufficient optimality conditions and constraint qualifications. We utilize the localization principle in order to obtain simple necessary and sufficient conditions for the global exactness of the extended penalty function introduced by Huyer and Neumaier, and in order to construct a globally exact continuously differentiable augmented Lagrangian function for nonlinear semidefinite programming problems.