Abstract
In this paper we give a survey of some recent results for quasidifferentiable functions and minimal representations of quasidifferentials. In particular, attention is paid to the problem of finding different types of minimal representatives of a pair of nonempty compact convex subsets of a locally convex topological vector space in terms of the Rådström -Hörmander Theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aubin, J.P. (1997), Mathematical Methods of Game and Economic Theory, North-Holland Publ. Comp. Amsterdam, New York, Oxford.
Bartels, S.G., Kuntz, L. and Scholtes, S. (1995), Continuous Selections of Linear Functions and Nonsmooth Critical Point Theory. Nonlinear Analysis, Theory, Meth. & AppL, 24, pp. 385–407.
Clarke, F.H. (1983), Optimization and Nonsmooth Analysis, John Wiley, New York.
Demyanov, V.F. and Rubinov, A.M. (1986), Quasidifferential Calculus, Optimization Software Inc., Publications Division, New York.
Demyanov, V.F. and Pallaschke, D. (1997), Point derivations for Lipschitz functions and Clarke’s generalized derivative. Applications Mathematicae, 24, pp. 465–474.
Ewald, G. (1996), Combinatorial Convexity and Algebraic Geometry, Springer Verlag, Berlin, Heidelberg, New York.
Grzybowski, J. (1994), Minimal pairs of compact convex sets, Archiv der Mathematik, 63, pp. 173–181.
Truong Xuan Due Ha, (1989), Banach spaces of d.c. functions and quasidifferentiable functions, Acta Math. Vietnam, 13, 2, pp. 55–70.
Hörmander, L. (1954), Sur la fonction d’ appui des ensembles convexes dans un espace localement convexe, Arkiv for Matematik, 3, pp. 181–186.
Klee, V. (1958), Extremal structure of convex sets II, Math. Zeitschrift 69, pp. 90–104.
McMullen, P. (1989), The Polytope Algebra. Advances in Mathematics 78, pp. 76–130.
Melzer, D. (1986), On the Expressibility of Piecewise-Linear Continuous Functions as the Difference of two Piecewise-Linear Convex Functions. Math. Programming Study 29, pp. 118 — 134.
Pallaschke, D. and Rolewicz, S. (1997), Foundations of Mathematical Optimization, Mathematics and its Applications, Kluwer Acad. Publ. Dordrecht.
Pallaschke, D., Scholtes, S. and Urbanski, R. (1991), On minimal pairs of compact convex sets, Bull. Polish Acad. Sei. Math. 39, pp. 1–5.
Pallaschke, D. and Urbanski, R. (1993), Some criteria for the minimality of pairs of compact convex sets, Zeitschrift für Operations Research (ZOR) 37, pp. 129–150.
Pallaschke, D. and Urbanski, R. (1994), Reduction of quasidifFerentials and minimal representations,Ma£/*em. Programming, (Series A) 66 pp. 161–180.
Pallaschke, D. and Urbanski, R. (1996), A continuum of minimal pairs of compact convex sets which are not connected by translations, Journal of Convex Analysis 3, pp. 83–95.
Pallaschke, D., Urbanska, W. and Urbanski, R. (1997), C-Minimal Pairs of Compact Convex Sets, Journal of Convex Analysis 4, pp. 1–26.
Phelps, R.R. (1989), Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, Vol. 1364, Springer-Verlag, Berlin, Heidelberg, New York.
Pinsker, A.G. (1966), The space of convex sets of a locally convex space, Trudy Leningrad Engineering-Economic Institute 63, pp. 13–17.
Radström, H. (1952), An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3, pp. 165–169.
Rockafellar, R.T (1970), Convex Analysis, Princeton University Press, Princeton, New Jersey.
Rolewicz, S. (1984), Metric Linear Spaces, PWN and D.Reidel Publ. Company, Warszawa-Dordrecht.
Rolewicz, S. (1987), Functional Analysis and Control Theory, PWN-Polish Scientific Publishers, Warszawa and D. Reidel Publishing Company, Dordrecht.
SaJlee, G.T. (1974), On the indecomposibility of the cone, Journ. London Math. Soc. 9, pp. 363–367.
Scholtes, S. (1992), Minimal pairs of convex bodies in two dimensions, Mathematika 39, pp. 267–273.
Sherbert, D.R. (1964), The structure of ideals and point derivations in Banach
Algebras of Lipschitz functions, Trans AMS 111, pp. 240–272.
Urbanski, R. (1976), A generalization of the Minkowski-Radström-Hörman der Theorem, Bull. Acad. Polon. Sei. Ser. Sei. Math. 288, pp. 709–715.
Urbanski, R. (1996), On minimal convex pairs of convex compact sets, Archiv der Mathematik 67, pp. 226–238.
Wiernowolski, M. (1994), On Amount of Minimal Pairs, Funct. Approximat. Comment. Math. 23, pp. 35–39.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Pallaschke, D., Urbański, R. (2000). Minimal Pairs of Compact Convex Sets, with Application to Quasidifferential Calculus. In: Demyanov, V., Rubinov, A. (eds) Quasidifferentiability and Related Topics. Nonconvex Optimization and Its Applications, vol 43. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3137-8_8
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3137-8_8
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4830-4
Online ISBN: 978-1-4757-3137-8
eBook Packages: Springer Book Archive