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Understanding the role that subgradients play in various second-ordervariational analysis constructions can help us uncover new properties ofimportant classes of functions in variational analysis. Focusing mainly on thebehavior of the second subderivative and subgradient proto-derivative ofpolyhedral functions, functions with polyhedral epigraphs, we demonstrate thatchoosing the underlying subgradient, utilized in the definitions of theseconcepts, from the relative interior of the subdifferential of polyhedralfunctions ensures stronger second-order variational properties such as stricttwice epi-differentiability and strict subgradient proto-differentiability.This allows us to characterize continuous differentiability of the proximalmapping and twice continuous differentiability of the Moreau envelope ofpolyhedral functions. We close the paper with proving the equivalence of metricregularity and strong metric regularity of a class of generalized equations attheir nondegenerate solutions.

Role of Subgradients in Variational Analysis of Polyhedral Functions