On relations and applications of generalized second-order directional derivatives

Relations among upper and lower generalized second-order directional derivatives for a locally Lipschitz function are derived. In addition, existence conditions are determined for second-order directional derivatives. The resulting relations and conditions are used to characterize the convexity property of a locally Lipschitz function and to compare generalized Hessians and second-order optimality conditions. Finally, it is demonstrated that a Cominetti-Correa-type directional derivative is bounded below by the conjugacy of a Ben-Tal-Zowe-type directional derivative.