A Notion of Fenchel Conjugate for Set-Valued Mappings

Nguyen Mau Nam (), Gary Sandine (), Nguyen Nang Thieu () and Nguyen Dong Yen ()
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Nguyen Mau Nam: Portland State University
Gary Sandine: Portland State University
Nguyen Nang Thieu: Vietnam Academy of Science and Technology
Nguyen Dong Yen: Vietnam Academy of Science and Technology

Journal of Optimization Theory and Applications, 2024, vol. 203, issue 2, No 8, 1263-1292

Abstract: Abstract In this paper, we present a novel concept of the Fenchel conjugate for set-valued mappings and investigate its properties in finite and infinite dimensions. After establishing some fundamental properties of the Fenchel conjugate for set-valued mappings, we derive its main calculus rules in various settings. Our approach is geometric and draws inspiration from the successful application of this method by B.S. Mordukhovich and coauthors in variational and convex analysis. Subsequently, we demonstrate that our new findings for the Fenchel conjugate of set-valued mappings can be utilized to obtain many old and new calculus rules of convex generalized differentiation in both finite and infinite dimensions.

Keywords: Fenchel conjugate; Coderivative; Subdifferential; Convex set-valued mapping; Relative interior; Quasi-relative interior; Strong quasi-relative interior; 49J52; 49J53; 90C31 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-024-02455-w

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