A pointed convex cone naturally induces a partial order, and further a notion of nondecreasingness for functions. We consider extended real-valued functions defined on the cone. Monotone conjugates for these functions can be defined in an analogous way to the standard convex conjugate. The only difference is that the supremum is taken over the cone instead of the entire space. We give sufficient conditions for the cone under which the corresponding Fenchel–Moreau biconjugation identity holds for proper, convex, lower semicontinuous, and nondecreasing functions defined on the cone. In addition, we show that these conditions are satisfied by a class of cones known as perfect cones.
Un cône convexe pointu induit naturellement un ordre partiel, et de plus une notion de non-décroissance pour les fonctions. Nous considérons des fonctions étendues à valeurs réelles définies sur le cône. Les conjugués monotones de ces fonctions peuvent être définis de manière analogue au conjugué convexe standard. La seule différence est que le supremum est pris sur le cône au lieu de l’espace entier. Nous donnons des conditions suffisantes pour le cône sous lesquelles l’identité de biconjugaison de Fenchel–Moreau correspondante a lieu pour les fonctions propres, convexes, semi-continues inférieures et non décroissantes définies sur le cône. En outre, nous montrons que ces conditions sont satisfaites par une classe de cônes connue sous le nom de cônes parfaits.
Accepté le :
Publié le :
DOI : 10.5802/afst.1771
Keywords: Fenchel–Moreau identity, monotone conjugate, cone
Chen, Hong-Bin 1 ; Xia, Jiaming 2
CC-BY 4.0
@article{AFST_2024_6_33_2_287_0,
author = {Chen, Hong-Bin and Xia, Jiaming},
title = {Fenchel{\textendash}Moreau identities on convex cones},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {287--309},
year = {2024},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 33},
number = {2},
doi = {10.5802/afst.1771},
mrnumber = {4806784},
zbl = {07938067},
language = {en},
url = {https://www.numdam.org/articles/10.5802/afst.1771/}
}
TY - JOUR AU - Chen, Hong-Bin AU - Xia, Jiaming TI - Fenchel–Moreau identities on convex cones JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2024 SP - 287 EP - 309 VL - 33 IS - 2 PB - Université Paul Sabatier, Toulouse UR - https://www.numdam.org/articles/10.5802/afst.1771/ DO - 10.5802/afst.1771 LA - en ID - AFST_2024_6_33_2_287_0 ER -
%0 Journal Article %A Chen, Hong-Bin %A Xia, Jiaming %T Fenchel–Moreau identities on convex cones %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2024 %P 287-309 %V 33 %N 2 %I Université Paul Sabatier, Toulouse %U https://www.numdam.org/articles/10.5802/afst.1771/ %R 10.5802/afst.1771 %G en %F AFST_2024_6_33_2_287_0
Chen, Hong-Bin; Xia, Jiaming. Fenchel–Moreau identities on convex cones. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 2, pp. 287-309. doi: 10.5802/afst.1771
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