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Journal of Convex Analysis, Vol. 4, No. 2, pp. 221-234 (1997)
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Proto-Derivatives of Partial Subgradient Mappings

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R. A. Poliquin and R. T. Rockafellar

Dept. of Mathematical Sciences, Univ. of Alberta, Edmonton, Alberta, T6G 2G1, Canada, rene.poliquin@ualberta.ca, and Dept. of Mathematics, Univ. of Washington, Seattle, WA 98195, USA, rtr@math.washington.edu

**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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