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On concepts of directional differentiability

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Abstract

Various definitions of directional derivatives in topological vector spaces are compared. Directional derivatives in the sense of Gâteaux, Fréchet, and Hadamard are singled out from the general framework of σ-directional differentiability. It is pointed out that, in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equivalent. The chain rule for directional derivatives of a composite mapping is discussed.

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Communicated by O. L. Mangasarian

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Shapiro, A. On concepts of directional differentiability. J Optim Theory Appl 66, 477–487 (1990). https://doi.org/10.1007/BF00940933

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