Abstract
In this paper we generalize the Demyanov difference to the case of real Hausdorff topological vector spaces. We prove some classical properties of the Demyanov difference. In the proofs we use a new technique which is based on the properties given in Lemma 1. Due to its importance it will be called the preparation lemma. Moreover, we give connections between Minkowski subtraction and the union of upper differences. We show that in the case of normed spaces the Demyanov difference coincides with classical definitions of Demyanov subtraction.
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References
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Demyanov, V.F., Rubinov, A.M.: Quasidifferential Calculus. Springer and Optimization Software Inc., New York (1986)
Hukuhara, M.: Intégration des applications mesurables dont la valeur est un compact convexe (French). Funkcial. Ekvac. 10, 205–223 (1967)
Husain, T., Tweddle, I.: On the extreme points of the sum of two compact convex sets. Math. Ann. 188, 113–122 (1970)
Klee, V.: Extremal structure of convex sets II. Math. Zeitschr. 69, 90–104 (1958)
Pallaschke, D., Przybycień, H., Urbański, R.: On partialy ordered semigroups. J. Set Valued Anal. 16, 257–265 (2007)
Pallaschke, D., Urbański, R.: Pairs of Compact Convex Sets: Fractional Arithmetic with Convex Sets, Mathematics and Its Applications. Kluwer Academic Publisher, Dortrecht (2002)
Pallaschke, D., Urbański, R.: On the separation and order law of cancellation for bounded sets. Optimization 51, 487–496 (2002)
Pinsker A.G.: The space of convex sets of a locally convex space. Trudy Leningrad Eng. Econ. Inst. 63, 13–17 (1966)
Pontryagin L.S.: On linear differential games II (Russian). Dokl. Acad. Nauk SSSR 175, 764–766 (1967)
Rubinov A.M., Akhundov, I.S.: Differences of compact sets in the sense of Demyanov and its application to non-smooth-analysis. Optimization 23, 179–189 (1992)
Urbański, R.: A generalization of the Minkowski–Rådström–Hörmander theorem. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24, 709–715 (1976)
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Grzybowski, J., Pallaschke, D., Urbański, R. (2014). Demyanov Difference in Infinite-Dimensional Spaces. In: Demyanov, V., Pardalos, P., Batsyn, M. (eds) Constructive Nonsmooth Analysis and Related Topics. Springer Optimization and Its Applications, vol 87. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8615-2_2
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DOI: https://doi.org/10.1007/978-1-4614-8615-2_2
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