ON LOWER BOUNDS OF THE 2ND-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 dire ction u, the upper second-order derivative D(+)(2)f(.;u, 0) (in the se nse of Dem'yanov and Pevnyi 1974; Ben-Tal and Zowe 1982) has the same lower bounds as the lower second-order derivatives D(-)(2)f(.; u, 0). Consequently, one can characterize the convexity of f in terms of thes e derivatives. We also obtain the corresponding results in terms of Ch aney's second-order directional derivatives.