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Role of Subgradients in Variational Analysis of Polyhedral Functions

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Abstract

Understanding the role that subgradients play in various second-order variational analysis constructions can help us uncover new properties of important classes of functions in variational analysis. Focusing mainly on the behavior of the second subderivative and subgradient proto-derivative of polyhedral functions, i.e., functions with polyhedral convex epigraphs, we demonstrate that choosing the underlying subgradient, utilized in the definitions of these concepts, from the relative interior of the subdifferential of polyhedral functions ensures stronger second-order variational properties such as strict twice epi-differentiability and strict subgradient proto-differentiability. This allows us to characterize continuous differentiability of the proximal mapping and twice continuous differentiability of the Moreau envelope of polyhedral functions. We close the paper with proving the equivalence of metric regularity and strong metric regularity of a class of generalized equations at their nondegenerate solutions.

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Acknowledgements

The first author would like to thank Vietnam Institute for Advanced Study in Mathematics for hospitality during her postdoctoral fellowship of the Institute in 2021–2022. Research of the first author is partially supported by Vietnam Academy of Science and Technology under grant CTTH00.01/22-23. Research of the third author is partially supported by the U.S. National Science Foundation under the Grant DMS 2108546.

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Authors and Affiliations

  1. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

    Nguyen T. V. Hang

  2. Department of Mathematics, Miami University, Oxford, OH, 45065, USA

    Woosuk Jung & Ebrahim Sarabi

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  1. Nguyen T. V. Hang
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  2. Woosuk Jung
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  3. Ebrahim Sarabi
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Correspondence to Ebrahim Sarabi.

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Hang, N.T.V., Jung, W. & Sarabi, E. Role of Subgradients in Variational Analysis of Polyhedral Functions. J Optim Theory Appl 200, 1160–1192 (2024). https://doi.org/10.1007/s10957-024-02378-6

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