Abstract
In this paper, we develop the sufficient conditions for the existence of local and global saddle points of two classes of augmented Lagrangian functions for nonconvex optimization problem with both equality and inequality constraints, which improve the corresponding results in available papers. The main feature of our sufficient condition for the existence of global saddle points is that we do not need the uniqueness of the optimal solution. Furthermore, we show that the existence of global saddle points is a necessary and sufficient condition for the exact penalty representation in the framework of augmented Lagrangians. Based on these, we convert a class of generalized semi-infinite programming problems into standard semi-infinite programming problems via augmented Lagrangians. Some new first-order optimality conditions are also discussed.
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This research was supported by the National Natural Science Foundation of P.R. China (Grant No. 10571106 and No. 10701047).
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Wang, C., Zhou, J. & Xu, X. Saddle Points Theory of Two Classes of Augmented Lagrangians and Its Applications to Generalized Semi-infinite Programming. Appl Math Optim 59, 413–434 (2009). https://doi.org/10.1007/s00245-008-9060-y
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DOI: https://doi.org/10.1007/s00245-008-9060-y