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Second-order characterization of convex mappings in Banach spaces and its applications

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Abstract

We show that the positive semi-definiteness of the regular or limiting (Mordukhovich) second-order subdifferential of an approximately convex function is a sufficient condition for its convexity. As a consequence of our result, we obtain a second-order characterization for the class of lower-\(C^1\) functions. Furthermore, we show by an example that positive semi-definiteness of the second-order subdifferential of convex functions is not a necessary condition for some cases. Also, a second-order characterization for C-convex mappings is obtained, and derive some applications in optimization.

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Acknowledgements

This research was partly supported by the Iranian National Science Foundation (INSF) under the contract No. 99014611. The authors are grateful to the anonymous reviewer for the careful reading of the manuscript and especially for the constructive comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Isfahan, Isfahan, 81745-163, Iran

    Mohammad Taghi Nadi

  2. Department of Mathematics, Sheikhbahaee University and University of Isfahan, Isfahan, Iran

    Jafar Zafarani

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  1. Mohammad Taghi Nadi
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  2. Jafar Zafarani
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Correspondence to Jafar Zafarani.

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Nadi, M.T., Zafarani, J. Second-order characterization of convex mappings in Banach spaces and its applications. J Glob Optim 86, 1005–1023 (2023). https://doi.org/10.1007/s10898-023-01301-z

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