Geodesic
The hypodifferential and the $\varepsilon$-subdifferential of polyhedral function
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2011), pp. 64-71
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The class of polyhedral functions is the simplest among the family of nonsmooth functions. One of the basic concepts of convex analysis is the notion of $\varepsilon$-subdifferential. The $\varepsilon$-subdifferential mapping is continuous in the Hausdorff metric. This property is used in the construction of continuous optimization methods for convex functions. The notions of hypodifferential and continuous hypodifferential were introduced by V. F. Demyanov. A polyhedron of special form can be taken as a continuous hypodifferential for a polyhedral function. In the paper, the properties of this hypodifferential and the $\varepsilon$-subdifferential of the polyhedral function are discussed. The relationship between them is established. A geometric interpretation of the hypodifferential is derived and the examples illustrating application of the developed theory are presented.
Keywords: convex function, subdifferential, hypodifferential. 
@article{VSPUI_2011_3_a7,
     author = {L. N. Polyakova},
     title = {The hypodifferential and the $\varepsilon$-subdifferential of polyhedral function},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {64--71},
     year = {2011},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2011_3_a7/}
}
TY  - JOUR
AU  - L. N. Polyakova
TI  - The hypodifferential and the $\varepsilon$-subdifferential of polyhedral function
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2011
SP  - 64
EP  - 71
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2011_3_a7/
LA  - ru
ID  - VSPUI_2011_3_a7
ER  - 
%0 Journal Article
%A L. N. Polyakova
%T The hypodifferential and the $\varepsilon$-subdifferential of polyhedral function
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2011
%P 64-71
%N 3
%U http://geodesic.mathdoc.fr/item/VSPUI_2011_3_a7/
%G ru
%F VSPUI_2011_3_a7
L. N. Polyakova. The hypodifferential and the $\varepsilon$-subdifferential of polyhedral function. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 3 (2011), pp. 64-71. http://geodesic.mathdoc.fr/item/VSPUI_2011_3_a7/

[1] Rokafellar R., Vypuklyi analiz, per. s angl. A. D. Ioffe, V. M. Tikhomirova, Mir, M., 1973, 469 pp.

[2] Nurminskii E. A., “O nepreryvnosti $\varepsilon$-subgradientnykh otobrazhenii”, Kibernetika, 1977, no. 5, 148–149

[3] Demyanov V. F., Vasilev L. V., Nedifferentsiruemaya optimizatsiya, Nauka, M., 1981, 383 pp.

[4] Kusraev A. G., Kutateladze S. S., Subdifferentsialnoe ischislenie, Nauka, Sib. otd., Novosibirsk, 1987, 224 pp.

[5] Demyanov V. F., Rubinov A. M., Osnovy negladkogo analiza i kvazidifferentsialnoe ischislenie, Nauka, M., 1990, 431 pp.

[6] Polyakova L. N., “Zadacha globalnoi optimizatsii raznosti poliedralnykh funktsii”, Vestn. S.-Petepb. un-ta. Sep. 10: Prikladnaya matematika, informatika, protsessy upravleniya, 2006, no. 4, 83–96