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 functions, or more generally as the composition of a piecewise linear-quadratic function with a mapping, is minimized subject to finitely many 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:
https://doi.org/10.1090/S0002-9947-1988-0936806-9

Keywords:
Nonlinear programming,
nonsmooth programming,
epi-convergence,
epi-derivatives,
generalized second derivatives

Article copyright:
© Copyright 1988
American Mathematical Society