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Directional Differentiability of the Metric Projection Operator in Uniformly Convex and Uniformly Smooth Banach Spaces

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Abstract

Let X be a real uniformly convex and uniformly smooth Banach space and C a nonempty closed and convex subset of X. Let PC: X → C denote the (standard) metric projection operator. In this paper, we define the G\(\widehat{a}\)teaux directional differentiability of PC. We investigate some properties of the G\(\widehat{a}\)teaux directional differentiability of PC. In particular, if C is a closed ball or a closed and convex cone (including proper closed subspaces), then, we give the exact representations of the directional derivatives of PC.

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Acknowledgements

The author is very grateful to Professors Phil Blau, Li Cheng, Akhtar Khan, Robert Mendris, and Preston Nichols for their kind communications in the development stage of this paper. The author deeply thanks Professors Dezhou Kong, Lishan Liu, Simeon Reich and Linsen Xie for their valuable comments and suggestions, which improved the presentation of this paper.

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  1. Department of Mathematics, Shawnee State University, Portsmouth, OH, 45662, USA

    Jinlu Li

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Correspondence to Jinlu Li.

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Communicated by Aviv Gibali.

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Li, J. Directional Differentiability of the Metric Projection Operator in Uniformly Convex and Uniformly Smooth Banach Spaces. J Optim Theory Appl 200, 923–950 (2024). https://doi.org/10.1007/s10957-023-02329-7

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