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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 may fail to be locally Lipschitz continuous. It covers a range of important, yet challenging, applications, including inverse optimal value optimization and problems under 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 it 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. To demonstrate that our algorithmic framework is practically implementable, we further present verifiable termination criteria and preliminary numerical results. Funding: Financial support from the National Science Foundation Division of Computing and Communication Foundations [Grant CCF-2416172] and Division of Mathematical Sciences [Grant DMS-2416250] and the National Cancer Institute, National Institutes of Health [Grant 1R01CA287413-01] is gratefully acknowledged. 

Abstract Geologic records support a short-lived carbon release, known as the pre-onset excursion (POE), shortly before the Paleocene-Eocene Thermal Maximum (PETM; ~ 56 Ma). However, the source and pace of the POE carbon release and its relationship to the PETM remain unresolved. Here we show a high-temporal-resolution stratigraphic record spanning the POE and PETM from the eastern Tethys Ocean that documents the evolution of surface ocean carbon cycle, redox and eutrophication, confirming the global nature of the POE. Biomarkers extracted from the sedimentary record indicate a smaller environmental perturbation during the POE than that during the PETM in the eastern Tethys Ocean. Earth system modeling constrained by observed δ¹³C and pH data indicates that the POE was driven by a largely thermogenic CO₂source, likely associated with sill intrusions prior to the main eruption phase of the North Atlantic Igneous Province and possibly biogeochemical feedbacks involving the release of biogenic methane. 

The minimization of nonlower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance constrained stochastic programs, a Heaviside composite optimization problem is one whose objective and constraints are defined by sums of possibly nonlinear multiples of such composite functions. Via a pulled-out formulation, a pseudostationarity concept for a feasible point was introduced in an earlier work as a necessary condition for a local minimizer of a Heaviside composite optimization problem. The present paper extends this previous study in several directions: (a) showing that pseudostationarity is implied by (and thus, weaker than) a sharper subdifferential-based stationarity condition that we term epistationarity; (b) introducing a set-theoretic sufficient condition, which we term a local convexity-like property, under which an epistationary point of a possibly nonlower semicontinuous optimization problem is a local minimizer; (c) providing several classes of Heaviside composite functions satisfying this local convexity-like property; (d) extending the epigraphical formulation of a nonnegative multiple of a Heaviside composite function to a lifted formulation for arbitrarily signed multiples of the Heaviside composite function, based on which we show that an epistationary solution of the given Heaviside composite program with broad classes of B-differentiable component functions can in principle be approximately computed by surrogation methods. Funding: The work of Y. Cui was based on research supported by the National Science Foundation [Grants CCF-2153352, DMS-2309729, and CCF-2416172] and the National Institutes of Health [Grant 1R01CA287413-01]. The work of J.-S. Pang was based on research supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0045].