Local Nonconvexity Indices for \(C^{1,1}\) Functions via Generalized Hessians

Abstract

Davydov, Moldavskaya, and Zitikis introduced local indices for quantifying the lack of convexity of a \(C^2\) function by measuring the nuclear-norm distance of its Hessian from the cone of positive semidefinite matrices. This paper develops a local analogue for functions of class \(C^{1,1}\). At a point \(x\), the classical Hessian is replaced by the Clarke-type generalized Hessian set \(\Hess(h;x)\), defined as the closed convex hull of limiting Hessians at nearby twice differentiability points. Evaluating the same spectral functional over \(\Hess(h;x)\) gives an interval-valued local nonconvexity index whose lower and upper endpoints represent, respectively, the least and greatest visible second-order nonconvexity at \(x\). The construction reduces to the original smooth index when \(h\in C^2\), vanishes for convex \(C^{1,1}\) functions, is invariant under orthogonal changes of variables, satisfies a subadditivity inequality for the upper endpoint under sums, and is upper semicontinuous in its upper endpoint. We also relate the upper endpoint to a pointwise weak-convexity curvature modulus and give explicit \(C^{1,1}\setminus C^2\) examples. The paper is deliberately local in scope: it proposes a scalar diagnostic extracted from generalized Hessian sets, not a replacement for the richer second-order variational theory of nonsmooth convexity.