Geodesic
Smooth approximations of nonsmooth convex functions
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 18 (2022) no. 4, pp. 535-547 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For an arbitrary convex function, using the infimal convolution operation, a family of continuously differentiable convex functions approximating it is constructed. The constructed approximating family of smooth convex functions Kuratowski converges to the function under consideration. If the domain of the considered function is compact, then such smooth convex approximations are uniform in the Chebyshev metric. The approximation of a convex set by a family of smooth convex sets is also considered.
Keywords: set-valued mapping, semicontinuous mapping, conjugate function, Kuratowski converge, smooth approximation.
Mots-clés : infimal convolution operation 
@article{VSPUI_2022_18_4_a7,
     author = {L. N. Polyakova},
     title = {Smooth approximations of nonsmooth convex functions},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {535--547},
     year = {2022},
     volume = {18},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2022_18_4_a7/}
}
TY  - JOUR
AU  - L. N. Polyakova
TI  - Smooth approximations of nonsmooth convex functions
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2022
SP  - 535
EP  - 547
VL  - 18
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2022_18_4_a7/
LA  - en
ID  - VSPUI_2022_18_4_a7
ER  - 
%0 Journal Article
%A L. N. Polyakova
%T Smooth approximations of nonsmooth convex functions
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2022
%P 535-547
%V 18
%N 4
%U http://geodesic.mathdoc.fr/item/VSPUI_2022_18_4_a7/
%G en
%F VSPUI_2022_18_4_a7
L. N. Polyakova. Smooth approximations of nonsmooth convex functions. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 18 (2022) no. 4, pp. 535-547. http://geodesic.mathdoc.fr/item/VSPUI_2022_18_4_a7/

[1] Bertsekas P., Convex optimization theory, Athena Scientific Publ, Belmont, MA, 2009, 444 pp. | MR

[2] Leichtweiß K., Konvexe mengen, Springer-Verlag Publ, Berlin, 1980, 330 pp. | MR

[3] Rockafellar R. T., Convex analysis, Princeton University Press, Princeton, 1970, 472 pp. | MR

[4] Zillober C., “Convex approximation methods for practical optimization”, Operations Research Proceedings, 2000, eds. B. Fleischmann, R. Lasch, U. Derigs, W. Domschke, U. Rieder, Springer Publ, Berlin–Heidelberg, 2001, 20–25 | DOI | MR

[5] Moreau J.-J., “Propriétés des applications “prox””, C. R. Acad. Sci. (Paris), 256 (1963), 1069–1071 | MR

[6] Moreau J.-J., “Proximité et dualité dans un espace Hilbertien”, Bull. Soc. Math. France, 93 (1965), 273–299 | DOI | MR

[7] Planiden C., Wang X., “Strongly convex functions. Moreau envelopes and the generic nature of convex functions with strong minimizers”, SIAM J. Optim., 26:2 (2016), 1341–1364 | DOI | MR

[8] Wijsman R. A., “Convergence of sequences of convex sets, cones, and functions. II”, Trans. Amer. Math. Soc., 123 (1966), 32–45 | DOI | MR

[9] Beer G., “On convergence of closed sets in a metric space and distance functions”, Bull. Australian Math. Soc., 31 (2009), 421–432 | DOI | MR

[10] Kuratowski K., Topologie, v. I, II, PWN Monogr. Mat., Panstwowe Wydawnictwa Naukowe Publ, Warszawa, 1961, 528 pp.

[11] Polyakova L. N., “On a smooth approximation of convex funchions”, VIII Moscow Internation Conference on Operations Research, ORM 2016, MAKS Press, M., 2016, 58–61 (In Russian)