  We investigate a class of composite nonconvex functions, where the outer
function is the sum of univariate extended-real-valued convex functions and the
inner function is the limit of difference-of-convex functions. A notable
feature of this class is that the inner function can be merely lower
semicontinuous instead of continuously differentiable. It covers a range of
important yet challenging applications, including the composite value functions
of nonlinear programs and the value-at-risk constraints. We propose an
asymptotic decomposition of the composite function that guarantees
epi-convergence to the original function, leading to necessary optimality
conditions for the corresponding minimization problem. The proposed
decomposition also enables us to design a numerical algorithm such that any
accumulation point of the generated sequence, if exists, satisfies the newly
introduced optimality conditions. These results expand on the study of
so-called amenable functions introduced by Poliquin and Rockafellar in 1992,
which are compositions of convex functions with smooth maps, and the
prox-linear methods for their minimization.
