Lower- and upper-C² integrands: a structured class of functions for stochastic nonconvex optimization

The authors extend lower-\(C^2\) functions, pointwise maxima of twice continuously differentiable functions over a compact index set, to normal integrands, where the index set is random and the Hessians are integrable and bounded. Their main result is that such integrands satisfy a local sufficient ascent property almost surely; equivalently, they differ from a convex normal integrand by a random quadratic. Upper-\(C^2\) integrands follow by a change of sign and appear as second-stage value functions of stochastic programs with compact recourse.

For this class, integrability of the function values, of the local Lipschitz moduli and of the minimum-norm Clarke subgradients are shown to be equivalent. The paper places smooth integrands, difference-of-convex integrands, Moreau envelopes and several compositions in the framework, and shows that the variational formulation of conditional value-at-risk fits within it, giving structure for stochastic methods on nonconvex risk objectives.