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Twice epi-differentiability of a class of non-amenable composite functions
This paper focuses on the twice epi-differentiability of a class of
non-amenable functions, which are the composition of a piecewise twice
differentiable (PWTD) function and a parabolically semidifferentiable mapping.
Such composite functions frequently appear those optimization problems covering
major classes of constrained and composite optimization problems, as well as
encompassing disjunctive programs and low-rank or/and sparsity optimization
problems. To achieve the goal, we first justify the proper twice
epi-differentiability, parabolic epi-differentiability and regularity of PWTD
functions, and derive an upper and lower estimate for the second subderivatives
of this class of composite functions in terms of a chain rule of their
parabolic subderivatives. Then, we employ the obtained upper and lower
estimates to characterize the parabolic regularity and second subderivatives
and then achieve the proper twice epi-differentiability for several classes of
popular functions, which include the compositions of PWTD outer functions and
twice differentiable inner mappings, the regularized functions inducing group
sparsity, and the indicator functions of the -order cone.Comment: 39page
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