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

Jinlu Li ()
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Jinlu Li: Shawnee State University

Journal of Optimization Theory and Applications, 2024, vol. 200, issue 3, No 2, 923-950

Abstract: 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 $$\widehat{a}$$ a ^ teaux directional differentiability of PC. We investigate some properties of the G $$\widehat{a}$$ 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.

Keywords: Uniformly convex and uniformly smooth Banach space; Metric projection operator; Directional differentiability of the metric projection; Directional derivative of the metric projection; 49J50; 26A24; 47A58; 47J30; 49J40 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-023-02329-7

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