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On generalized convex functions and generalized subdifferential

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Abstract

We introduce and study the notion of \(\sigma \)-subdifferential of a proper function f which contains the Clarke–Rockafellar subdifferential of f under some mild assumptions on f and \(\sigma \). We show that some well known properties of the convex function, namely Lipschitz property on the interior of its domain, remain valid for the large class of \(\sigma \)-convex functions.

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Notes

  1. We may take for example A to be the ball \(\left\{ x\in \mathbb {R}^{n}:\left\| x-x_{0}\right\| _{\infty }<\varepsilon \right\} \) with respect to the sup norm, and \(x_{i}\) to be its vertices.

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Acknowledgements

The author expresses his gratitude to Professor Nicolas Hadjisavvas for his many insightful comments, suggestions, and discussions. Also, I would like to thank the two Referees for their valuable comments on the manuscript and their suggestions and corrections for improving the document.

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Authors and Affiliations

  1. Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), 45137-66781, Zanjan, Iran

    Mohammad Hossein Alizadeh

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  1. Mohammad Hossein Alizadeh
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Correspondence to Mohammad Hossein Alizadeh.

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Alizadeh, M.H. On generalized convex functions and generalized subdifferential. Optim Lett 14, 157–169 (2020). https://doi.org/10.1007/s11590-018-1326-y

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  • DOI: https://doi.org/10.1007/s11590-018-1326-y

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