Journal of Convex Analysis, Vol. 4, No. 2, pp. 221-234 (1997)

Proto-Derivatives of Partial Subgradient Mappings

R. A. Poliquin and R. T. Rockafellar

Dept. of Mathematical Sciences, Univ. of Alberta, Edmonton, Alberta, T6G 2G1, Canada,, and Dept. of Mathematics, Univ. of Washington, Seattle, WA 98195, USA,

Abstract: Partial subgradient mappings have a key role in the sensitivity analysis of first-order conditions for optimality, and their generalized derivatives are especially important in that respect. It is known that such a mapping is proto-differentiable when it comes from a fully amenable function with compatible parameterization, which is a common case in applications; the proto-derivatives can be evaluated then through projections. Here this result is extended to a still broader class of functions than fully amenable, namely, ones obtained by composing a $C^2$ mapping with a kind of piecewise-$C^2$ convex function under a constraint qualification.

Keywords: Variational analysis, subgradient mappings, proto-derivatives, second-order epi-derivatives, amenable functions, piecewise-$C^2$ functions, nonsmooth analysis

Classification (MSC2000): 49A52, 58C06, 58C20; 90C30

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