  This paper concerns the twice epi-differentiability and parabolic regularity
of a class of non-amenable functions, the composition of a piecewise twice
differentiable (PWTD) function and a parabolically semidifferentiable mapping.
Such composite functions often appear in composite optimization problems,
disjunctive optimization problems, and low-rank and/or sparsity optimization
problems. By establishing the proper twice epi-differentiability and parabolic
epi-differentiability of PWTD functions, we prove the parabolic
epi-differentiability of this class of composite functions, and its twice
epi-differentiability under the parabolic regularity assumption. Then, we
identify a condition to ensure its parabolic regularity with the help of an
upper and lower estimate of its second subderivative, and demonstrate that this
condition holds for several classes of specific non-amenable functions.
