Generalized second derivatives of convex functions and saddle functions
R. TYRRELL ROCKAFELLAR · Transactions of the American Mathematical Society · 1990
The theory of second-order epi-derivatives of extended-real-valued functions is applied to convex functions on R n {\mathbb {R}^n} and shown to be closely tied to proto-differentiation of the corresponding subgradient multifunctions, as well as to second-order epi-differentiation of conjugate functions. An extension is then made to saddle functions, which by definition are convex in one argument and concave in another. For this case a concept of epi-hypo-differentiability is introduced. The saddle function results provide a foundation for the sensitivity analysis of primal and dual optimal solutions to general finite-dimensional problems in convex optimization, since such solutions are characterized as saddlepoints of a convex-concave Lagrangian function, or equivalently as subgradients of the saddle function conjugate to the Lagrangian.