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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

First- and second-order epi-differentiability in nonlinear programming


Author: R. T. Rockafellar
Journal: Trans. Amer. Math. Soc. 307 (1988), 75-108
MSC: Primary 90C48; Secondary 49A52, 58C20, 90C30
MathSciNet review: 936806
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Abstract: Problems are considered in which an objective function expressible as a max of finitely many $ {C^2}$ functions, or more generally as the composition of a piecewise linear-quadratic function with a $ {C^2}$ mapping, is minimized subject to finitely many $ {C^2}$ constraints. The essential objective function in such a problem, which is the sum of the given objective and the indicator of the constraints, is shown to be twice epi-differentiable at any point where the active constraints (if any) satisfy the Mangasarian-Fromovitz qualification. The epi-derivatives are defined by taking epigraphical limits of classical first-and second-order difference quotients instead of pointwise limits, and they reveal properties of local geometric approximation that have not previously been observed.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0936806-9
PII: S 0002-9947(1988)0936806-9
Keywords: Nonlinear programming, nonsmooth programming, epi-convergence, epi-derivatives, generalized second derivatives
Article copyright: © Copyright 1988 American Mathematical Society