Problems of nonlinear programming are placed in a broader framework of composite optimization. This allows second-order smoothness in the data structure to be utilized despite apparent nonsmoothness in the objective. Second-order epi-derivatives are shown to exist as expressions of such underlying smoothness, and their connection with several kinds of second-order approximation is examined. Expansions of the Moreau envelope functions and proximal mappings associated with the essential objective functions for certain optimization problems in composite format are studied in particular
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