Geodesic
The quasidifferential calculus, separation of convex sets and the Demyanov difference
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 14 (2018) no. 1, pp. 20-30
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In this paper we give a survey of the influence of the quasidifferential calculus of V. F. Demyanov and A. M. Rubinov to the field of generalized convexity. In particular, we will show the strong relations between the order cancellation property of bounded closed convex set and the separation property of bounded closed convex sets by sets. Moreover, a generalization of the Demyanov difference of compact convex sets infinite dimension and its role in the context of convex sets by sets is discussed. Refs 15. Figs 2.
Keywords: quasidifferential calculus, convex sets, subdifferential calculus, generaliz convexity. 
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J. Grzybowski; D. Pallaschke; R. Urbański. The quasidifferential calculus, separation of convex sets and the Demyanov difference. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 14 (2018) no. 1, pp. 20-30. http://geodesic.mathdoc.fr/item/VSPUI_2018_14_1_a2/

[1] Demyanov V. F., Rubinov A. M., “Quasidifferentiable functionals”, Papers of Academy of Sciences USSR, 250:1 (1980), 21–25 (In Russian) | MR | Zbl

[2] Demyanov V. F., Rubinov A. M., Quasidifferential calculus, Optimization Software Inc. Publ. Division, New York, 1986, 288 pp. | MR | Zbl

[3] Grzybowski J., Pallaschke D., Urbański R., “A pre-classification and the separation law for closed bounded convex sets”, Optimization Methods and Software, 20 (2005), 219–229 | DOI | MR | Zbl

[4] Grzybowski J., Pallaschke D., Urbański R., “Demyanov difference in infinite dimensional spaces”, Proceedings of CNSA (Saint Petersburg, 2014), Springer Optim. Appl., 87, 2014, 13–24 | MR | Zbl

[5] Pallaschke D., Urbański R., “Pairs of compact convex sets — Fractional arithmetic with convex sets”, Mathematics and its Applications, 548 (2002), 487–496 | MR

[6] Pallaschke D., Rolewicz S., “Foundations of mathematical optimization”, Mathematics and its Applications, 426 (1997), 417–429 | MR

[7] Diamond P., Kloeden P., Rubinov A. M., Vladimirov A., “Comperative properties of three metrics in the space of compact convex sets”, Set-Valued Analysis, 5:3 (1997), 267–289 | DOI | MR | Zbl

[8] Hörmander L., “On the function of support sets convex in a space that is locally convex”, Archives for Mathematics, 3 (1954), 181–186

[9] Pinsker A. G., “The space of convex sets of a locally convex space”, Trudy Leningrad Engineering-Economic Institute, 63, 1966, 13–17 | MR

[10] Rubinov A. M., “Abstract convexity, global optimization and data classification”, Opsearch, 38 (2001), 247–265 | DOI | MR | Zbl

[11] Urbański R., “A generalization of the Minkowski—Rådström—Hörmander theorem”, Bull. Acad. Polon. Sci. Series Sci. Math. Astr. Phys., 24 (1976), 709–715 | MR | Zbl

[12] Rubinov A. M., Akhundov I. S., “Differences of compact sets in the sense of Demyanov and its application to non-smooth-analysis”, Optimization, 23 (1992), 179–189 | DOI | MR

[13] Clarke F. H., Optimization and Nonsmooth Analysis, J. Wiley Comp. Publ., New York, 1983, 306 pp. | MR | Zbl

[14] Husain T., Tweddle I., “On the extreme points of the sum of two compact convex sets”, Math. Ann., 188 (1970), 113–122 | DOI | MR | Zbl

[15] Gaudioso M., Gorgone E., Pallaschke D., “Separation of convex sets by Clarke subdifferential”, Optimization, 59 (2011), 1199–1210 | DOI | MR