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

Rockafellar, R. T.

doi:10.1090/S0002-9947-1988-0936806-9

First- and second-order epi-differentiability in nonlinear programming
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Trans. Amer. Math. Soc. 307 (1988), 75-108 Request permission

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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