A Comparison of Second-Order Epiderivatives: Calculus and Optimality Conditions

The first-order theory of generalized derivatives is now well developed. Significant progress toward a second-order theory has also been made. Two types of second-order directional derivatives appear to be particularly promising for applications to optimization: parabolic epiderivatives, which are well suited for necessary optimality conditions and satisfy standard calculus rules for a large class of functions; and the second-order epiderivatives of Rockafellar, which lead to general sufficient optimality conditions that are close to being necessary. In this paper, the properties of these two types of epiderivatives are surveyed and contrasted. Special results for C1,1 functions are examined in detail.