Twice epi-differentiablity and parabolic regularity of a class of non-amenable functions

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.