On Lower Bounds of the Second-Order Directional Derivatives of Ben-Tal, Zowe, and Chaney

Let f be a regular, locally Lipschitz real-valued function defined on an open convex subset of a normed space. We show that at any unit direction u, the upper second-order derivative D₊²f(·; u, 0) (in the sense of Dem'yanov and Pevnyi [Dem'yanov, V. F., A. B. Pevnyi. 1974. Expansion with respect to a parameter of the extremal values of game problems. USSR Computational Math. and Math. Phys.14 33–45.]; Ben-Tal and Zowe [Ben-Tal, A., J. Zowe. 1982. Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems. Math. Programming24 70–91.]) has the same lower bounds as the lower second-order derivatives D₋²f(·; u, 0). Consequently, one can characterize the convexity of f in terms of these derivatives. We also obtain the corresponding results in terms of Chaney's second-order directional derivatives.