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This paper concerns the twice epi-differentiability and parabolic regularityof a class of non-amenable functions, the composition of a piecewise twicedifferentiable (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 optimizationproblems. By establishing the proper twice epi-differentiability and parabolicepi-differentiability of PWTD functions, we prove the parabolicepi-differentiability of this class of composite functions, and its twiceepi-differentiability under the parabolic regularity assumption. Then, weidentify a condition to ensure its parabolic regularity with the help of anupper and lower estimate of its second subderivative, and demonstrate that thiscondition holds for several classes of specific non-amenable functions.

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