Set-Valued Analysis

, Volume 2, Issue 1–2, pp 275–290

Proto-derivative formulas for basic subgradient mappings in mathematical programming

  • R. A. Poliquin
  • R. T. Rockafellar

DOI: 10.1007/BF01027106

Cite this article as:
Poliquin, R.A. & Rockafellar, R.T. Set-Valued Anal (1994) 2: 275. doi:10.1007/BF01027106


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 manyC2 functions, and the subgradients of the indicator of a set defined by finitely manyC2 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 

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • R. A. Poliquin
    • 1
  • R. T. Rockafellar
    • 2
  1. 1.Department of MathematicsUniversity of AlbertaEdmontonCanada
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA

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