Proto-derivative formulas for basic subgradient mappings in mathematical programming R. A. Poliquin R. T. Rockafellar Article

Received: 01 December 1993 DOI :
10.1007/BF01027106

Cite this article as: Poliquin, R.A. & Rockafellar, R.T. Set-Valued Anal (1994) 2: 275. doi:10.1007/BF01027106
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Abstract Subgradient mappings associated with various convex and nonconvex functions are a vehicle for stating optimality conditions, and their proto-differentiability plays a role therefore in the sensitivity analysis of solutions to problems of optimization. Examples of special interest are the subgradients of the max of finitely manyC ^{2} functions, and the subgradients of the indicator of a set defined by finitely manyC ^{2} constraints satisfying a basic constraint qualification. In both cases the function has a property called full amenability, so the general theory of existence and calculus of proto-derivatives of subgradient mappings associated with fully amenable functions is applicable. This paper works out the details for such examples. A formula of Auslender and Cominetti in the case of a max function is improved in particular.

Mathematics Subjects Classifications (1991) Primary 49J52, 58C06, 58C20 Secondary 90C30

Key words Proto-derivatives generalized second derivatives nonsmooth analysis epi-derivatives subgradient mappings amenable functions This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983 for the first author and by the National Science Foundation under grant DMS-9200303 for the second author.

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Authors and Affiliations R. A. Poliquin R. T. Rockafellar 1. Department of Mathematics University of Alberta Edmonton Canada 2. Department of Mathematics University of Washington Seattle USA