Gateaux Differentiability Revisited

Abstract

We revisit some basic concepts and ideas of the classical differential calculus and convex analysis extending them to a broader frame. We reformulate and generalize the notion of Gateaux differentiability and propose new notions of generalized derivative and generalized subdifferential in an arbitrary topological vector space. Meaningful examples preserving the key properties of the original notion of derivative are provided.

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Notes

  1. 1.

    If f is convex and lower semicontinuous, core(\(\mathop {\hbox {dom}}f\)) coincides with the interior of \(\mathop {\hbox {dom}}f\); see, for instance, [2].

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Acknowledgements

The research was supported by the Australian Research Council, project DP160100854. The authors would like to thank Prof. Jean-Baptiste Hiriart Urruty for attracting their attention to the papers by Laurent Matziak [7, 8].

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Correspondence to Michel Théra.

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Abbasi, M., Kruger, A.Y. & Théra, M. Gateaux Differentiability Revisited. Appl Math Optim (2021). https://doi.org/10.1007/s00245-021-09754-y

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Keywords

  • Gateaux differentiability
  • Moreau–Rockafellar subdifferential
  • Convex function
  • Directional derivative

Mathematics Subject Classification

  • 49J52
  • 49J53
  • 90C30