A Notion of Fenchel Conjugate for Set-Valued Mappings

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 s...

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Bibliographic Details
Published in:Journal of optimization theory and applications Vol. 203; no. 2; pp. 1263 - 1292
Main Authors: Nam, Nguyen Mau, Sandine, Gary, Thieu, Nguyen Nang, Yen, Nguyen Dong
Format: Journal Article
Language:English
Published: New York Springer US 01.11.2024
Subjects:
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Engineering
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Theory of Computation
90C31
49J53
49J52
Coderivative
Subdifferential
Convex set-valued mapping
Relative interior
Strong quasi-relative interior
Fenchel conjugate
Quasi-relative interior
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Summary: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.
ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-024-02455-w