Lower- and upper-C2 integrands: a structured class of functions for stochastic nonconvex optimization
Abstract
We study nonconvex stochastic functions that admit a structured second-order representation, a generalization of lower- and upper-C2 functions to normal integrands. Similarly to the deterministic case, lower-C2 normal integrands are defined as the pointwise maxima over a compact random index set of functions that are twice-continuously differentiable in the decision variable with an integrable bounded Hessian. Our main result is a local value function characterization of this class of functions, namely, that they satisfy a local sufficient ascent property almost surely. Equivalently, they differ from a convex normal integrand by a random quadratic. The upper-C2 theory follows by a change of sign, and delivers the corresponding lower envelopes, the form in which these integrands appear as second-stage value functions of stochastic programs with compact recourse. We also show that for this class integrability of the function values, of the local Lipschitz moduli, and of the minimum-norm Clarke subgradients are equivalent. We then study smooth integrands, finite lower envelopes, difference-of-convex integrands, Moreau and forward--backward envelopes, compositions with a smooth map, monotone convex compositions, and compact-recourse stochastic programs within these classes of nonconvex stochastic functions. As an application, we show that the optimistic loss of partial-label regression and the variational formulation of the conditional value-at-risk fit within this framework.
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BibTeX
@misc{atenas2026lower,
title = {{Lower- and upper-$C^2$ integrands: a structured class of functions for stochastic nonconvex optimization}},
author = {Felipe Atenas and Alejandro Jofré and Pedro Pérez-Aros and David Torregrosa-Belén},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37629},
eprint = {2610.37629},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37629},
}
RIS
TY - UNPB
TI - Lower- and upper-C^2 integrands: a structured class of functions for stochastic nonconvex optimization
AU - Felipe Atenas
AU - Alejandro Jofré
AU - Pedro Pérez-Aros
AU - David Torregrosa-Belén
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37629
UR - https://arxiv.org/abs/2610.37629
AB - We study nonconvex stochastic functions that admit a structured second-order representation, a generalization of lower- and upper-C^2 functions to normal integrands. Similarly to the deterministic case, lower-C^2 normal integrands are defined as the pointwise maxima over a compact random index set of functions that are twice-continuously differentiable in the decision variable with an integrable bounded Hessian. Our main result is a local value function characterization of this class of functions, namely, that they satisfy a local sufficient ascent property almost surely. Equivalently, they differ from a convex normal integrand by a random quadratic. The upper-C^2 theory follows by a change of sign, and delivers the corresponding lower envelopes, the form in which these integrands appear as second-stage value functions of stochastic programs with compact recourse. We also show that for this class integrability of the function values, of the local Lipschitz moduli, and of the minimum-norm Clarke subgradients are equivalent. We then study smooth integrands, finite lower envelopes, difference-of-convex integrands, Moreau and forward--backward envelopes, compositions with a smooth map, monotone convex compositions, and compact-recourse stochastic programs within these classes of nonconvex stochastic functions. As an application, we show that the optimistic loss of partial-label regression and the variational formulation of the conditional value-at-risk fit within this framework.
ER -