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Quasidifferentiable functions and minimal pairs of compact convex sets

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Abstrakty
EN
 Abstract: We give a survey on some recent results for quasidifferentiable functions. Beside some general properties of this type of functions, critical points and the problem of local linearization in a neighborhood of a regular point are discussed. Moreover, the structure of the quasidifferential is investigated. In particular, attention is paid to the problem of finding minimal representatives of a pair of nonempty compact convex subsets of a locally convex topological vector space in the sense of the Rådström-Hörmander theory.
Twórcy
  • Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany
  • Wydział Matematyki i Informatyki, Uniwersytet im. Adama Mickiewicza, Matejki 48/49, PL-60-769 Poznań, Poland
Strony
Bibliografia
[1] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983, pp. 133-139.
[2] V. F. Demyanov and A. M. Rubinov, Quasidifferential Calculus, Optimization Software Inc., Publications Division, New York, 1986.
[3] J. Grzybowski, Minimal pairs of compact convex sets, Arch. Math., submitted.
[4] H. Halkin, Implicit functions and optimization problems without continuous differentiability of data, SIAM J. Control 12 (1974), 229-236.
[5] L. Hörmander, Sur la fonction d'appui des ensembles convexes dans un espace localement convexe, Ark. Mat. 3 (1954), 181-186.
[6] V. Klee, Extremal structure of convex sets II, Math. Z. 69 (1958), 90-104.
[7] L. Kuntz, Invertierbarkeit und implizite Auflösung quasidifferenzierbarer Funktionen, Mathematical Systems in Economics 127, Anton Hain Verlag, Frankfurt/M., 1992.
[8] N. Newns and A. Walker, Tangent planes to differentiable manifolds, J. London Math. Soc. 31 (1956), 400-407.
[9] D. Pallaschke, P. Recht and R. Urbański, On locally Lipschitz quasidifferentiable functions in Banach spaces, Optimization 17 (1986), 287-295.
[10] D. Pallaschke, S. Scholtes and R. Urbański, On minimal pairs of compact convex sets, Bull. Polish Acad. Sci. Math. 39 (1991), 1-5.
[11] D. Pallaschke and R. Urbański, Some criteria for the minimality of pairs of compact convex sets, Z. Operations Research 37 (1993), 129-150.
[12] D. Pallaschke and R. Urbański, Reduction of quasidifferentials and minimal representations, Math. Programming Ser. A (1994).
[13] A. G. Pinsker, The space of convex sets of a locally convex space, Trudy Leningrad Engineering-Economic Institute 63 (1966), 13-17.
[14] H. Rådström, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165-169.
[15] S. Rolewicz, Metric Linear Spaces, PWN and D. Reidel, Warszawa-Dordrecht, 1984.
[16] S. Scholtes, Minimal pairs of convex bodies in two dimensions, Mathematika 39 (1992), 267-273.
[17] R. Urbański, A generalization of the Minkowski-Rådström-Hörmander Theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math. 24 (1976), 709-715.
Kolekcja
DML-PL
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