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Gateaux Differentiability Revisited

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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. 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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Authors and Affiliations

  1. Department of Mathematics, University of Isfahan, Isfahan, Iran

    Malek Abbasi

  2. School of Engineering, Information Technology and Physical Science, Federation University Australia, Ballarat, VIC, Australia

    Alexander Y. Kruger

  3. XLIM UMR-CNRS 7252, Université de Limoges, Limoges, France

    Michel Théra

  4. Federation University Australia, Ballarat, VIC, Australia

    Michel Théra

Authors
  1. Malek Abbasi
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  2. Alexander Y. Kruger
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  3. Michel Théra
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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 84, 3499–3516 (2021). https://doi.org/10.1007/s00245-021-09754-y

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