Derivative Tests for Prox-Regularity and the Modulus of Convexity

Prox-regularity is a fundamental property of functions in second-order variational analysis that has come to be understood as marking the boundary for convexity-like behavior. Broad classes of examples have long been familiar, but there has not been any pointwise test based on generalized second derivatives to check for its presence. Such tests are developed here in terms of newly defined strict second subderivatives and strict second-order subdifferentials. They also identify the exact associated level of variational s-convexity, whether positive or negative. A recent criterion of Gfrerer for the level of variational s-convexity of a function already identified as prox-regular is shown to follow also from the theory of generalized twice differentiability of convex functions and their conjugates. Tests of the strong variational sufficient condition for local optimality are obtained as an application.