{'fraction_non_alphanumeric': 0.02097130242825607, 'fraction_numerical': 0.0, 'mean_word_length': 5.525179856115108, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Generalized second derivatives of convex functions and saddle functions

The theory of second-order epi-derivatives of extended-real-valued functions is applied to convex functions on Rin 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.