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Quasidifferentiable Calculus and Minimal Pairs of Compact Convex Sets

Publication date: 20.12.2012

Schedae Informaticae, 2012, Volume 21, pp. 107 - 125

https://doi.org/10.4467/20838476SI.12.007.0817

Authors

,
Diethard Pallaschke
University of Karlsruhe, Karlsruhe, Germany
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Ryszard Urbański
Adam Mickiewicz University in Poznań, Wieniawskiego 1, 61-712 Poznań, Poland
All publications →

Titles

Quasidifferentiable Calculus and Minimal Pairs of Compact Convex Sets

Abstract

The quasidifferential calculus developed by V.F. Demyanov and A.M. Rubinov provides a complete analogon to the classical calculus of differentiation for a wide class of nonsmooth functions. Although this looks at the first glance as a generalized subgradient calculus for pairs of subdifferentials it turns out that, after a more detailed analysis, the quasidifferential calculus is a kind of Fréchet-differentiation whose gradients are elements of a suitable Minkowski–Rådström–Hörmander space. One aim of the paper is to point out this fact. The main results in this direction are Theorem 1 and Theorem 5. Since the elements of the Minkowski–Rådström–Hörmander space are not uniquely determined, we focus our attention in the second part of the paper to smallest possible representations of quasidifferentials, i.e. to minimal representations. Here the main results are two necessary minimality criteria, which are stated in Theorem 9 and Theorem 11.
 

References

Bartels S.G., Kuntz L., Scholtes S.; Continuous selections of linear functions and nonsmooth critical point theory, Nonlinear Analysis, Theory, Meth. & Appl. 24, 1995, pp. 385–407.

Bauer Chr.; Minimal and reduced pairs of convex bodies, Geom. Dedicata 62, 1996, pp. 179–192.

Demyanov V.F., Rubinov A.M.; Quasidifferential Calculus, Optimization Software, Inc. Publications Division, New York 1986.

Ewald G.; Combinatorial Convexity and Algebraic Geometry, Springer Verlag, Berlin, Heidelberg, New York 1996.

Grzybowski J., Pallaschke D., Urba´nski R.; On the reduction of pairs of bounded closed convex sets, Studia Mathematica 189, 2008, pp. 1–12.

Grzybowski J., Pallaschke D., Urba´nski R.; Minimal pairs of bounded closed convex sets as minimal representations of elements of the Minkowski–R°adstr¨om–H¨ormander spaces, Banach Center Publ. 84, 2009, 31–55.

Grzybowski J., Pallaschke D., Przybycie´n H., Urba´nski R.; Commutative Semigroups with Cancellation Law: A Representation Theorem, Semigroup Forum 83, 2011, pp. 447–456.

Grzybowski J., Urba´nski R.; Minimal pairs of bounded closed convex sets, Studia Mathematica 126, 1997, pp. 95–99.

H¨ormander L.; Sur la fonction d’ appui des ensembles convexes dans un espace localement convexe, Arkiv f¨or Matematik 3, 1954, pp. 181–186.

Knyazeva M., Panina G.; An illustrated theory of hyperbolic virtual polytopes, Central European Journal of Mathematics 6, 2008, pp. 204–217.

K¨othe G.; Topological vector spaces. I., Translated from the German Die Grundlehren der mathematischen Wissenschaften, Bd. 159, Springer-Verlag, New York 1969.

Langevin R., Levitt G., Rosenberg H.; H´erissons et multih´erissons (enveloppes parametr´ees par leur application de Gauss), Singularities, Warszawa 1985, Banach Center Publ. 20, 1988, pp. 245–253.

Pallaschke D., Rolewicz S.; Foundations of Mathematical Optimization – Convex Analysis without Linearity, Mathematics and its Applications, Kluwer Acad. Publ., Dordrecht 1997.

Pallaschke D., Urba´nski R.; Pairs of Compact Convex Sets – Fractional Arithmetic with Convex Sets, Mathematics and its Applications, Kluwer Acad. Publ., Dordrecht 2002.

Panina G.; Rigidity and flexibility of virtual polytopes, Central European Journal of Mathematics 2, 2003, pp. 157–168.

Pinsker A.G.; The space of convex sets of a locally convex space, Trudy Leningrad Engineering-Economic Institute 63, 1966, pp. 13–17.

Pukhlikov A.V., Khovanski˘ı A.G.; Finitely additive measures of virtual polyhedra (Russian), Algebra i Analiz 4, 1992, pp. 161–185, translation in St. Petersburg Math. Journ. 4, 1993, pp. 337–356.

Rolewicz S.; An untypical example of a quasidifferentiable function, Oral communication at the Oberwolfach Conference on ’Operations Research’, Feb. 26–March 03, 1984.

Urbański R.; A generalization of the Minkowski-R°adstr¨om-H¨ormander theorem, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astr. Phys. 24, 1976, pp. 709–715.

Information

Information: Schedae Informaticae, 2012, Volume 21, pp. 107 - 125

Article type: Original article

Titles:

Polish:

Quasidifferentiable Calculus and Minimal Pairs of Compact Convex Sets

English:

Quasidifferentiable Calculus and Minimal Pairs of Compact Convex Sets

Authors

University of Karlsruhe, Karlsruhe, Germany

Adam Mickiewicz University in Poznań, Wieniawskiego 1, 61-712 Poznań, Poland

Published at: 20.12.2012

Article status: Open

Licence: None

Percentage share of authors:

Diethard Pallaschke (Author) - 50%
Ryszard Urbański (Author) - 50%

Article corrections:

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Publication languages:

English