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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Generalized second derivatives of convex functions and saddle functions

Author: R. T. Rockafellar
Journal: Trans. Amer. Math. Soc. 322 (1990), 51-77
MSC: Primary 90C30; Secondary 49J52
MathSciNet review: 1031242
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Abstract: The theory of second-order epi-derivatives of extended-real-valued functions is applied to convex functions on $ {\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.

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Keywords: Nonsmooth analysis, epi-derivatives, sensitivity analysis, saddle functions, subgradients, epi-hypo-derivatives
Article copyright: © Copyright 1990 American Mathematical Society