Generalized second-order directional derivatives and optimization with C11 functions

In this paper, a new generalized second-order directional derivative and a set-valued generalized Hessian are introudced for C1,1 functions in real Banach spaces. It is shown that this set-valued generalized Hessian is single-valued at a point if and only if the function is twice weakly Gateaux differentiable at the point and that the generalized second-order directional derivative is upper semi-continuous under a regularity condition. Various generalized calculus rules are also given for C1,1 functions. The generalized second-order directional derivative is applied to derive second-order necessary optimality conditions for mathematical programming problems.