Directed Derivatives of Convex Compact-Valued Mappings

  • Robert Baier
  • Elza M. Farkhi
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 54)


Convex compact sets can be embedded into the Banach space of directed sets. Directed sets allow a visualization as possibly non-convex, compact sets in ℝ n and hence, this space could be used to visualize differences of embedded convex compact sets. The main application is the visualization as well as the theoretical and numerical calculation of set-valued derivatives. Known notions of affine, semi-affine and quasi-affine maps and their derivatives are studied.


Directed sets Set-valued derivatives Differences of convex sets and their visualization Affine Semi-affine Quasi-affine maps Embedding of convex compact sets into a vector space Directed intervals 


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  1. [1]
    R. Baier and E. Farkhi. Directed sets and differences of convex compact sets. In Systems modelling and optimization, Proceedings of the 18 th IFIP TC7 Conference held in Detroit, Michigan, July 22–25, 1997,pages 135–143, Boca Raton et al., 1999. Chapman and Hall/CRC.Google Scholar
  2. [2]
    R. Baier and E. Farkhi. Differences of Convex Compact Sets in the Space of Directed Sets-Part I:The Space of Directed Sets. Set-Valued Anal., to appear:1–31, 2000.Google Scholar
  3. [3]
    R. Baier and E. Farkhi. Differences of Convex Compact Sets in the Space of Directed Sets-Part II: Visualization of Directed Sets. Set-Valued Anal., to appear:1–28, 2000.Google Scholar
  4. [4]
    H. Hadwiger. Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt. Math. Z., 53(3):210–218, 1950.MathSciNetCrossRefGoogle Scholar
  5. [5]
    P. L. Hörmander. Sur la fonction d’appui des ensembles convexes dans un espace localement convexe. Ark. Mat., 3(12):181–186, 1954.Google Scholar
  6. [6]
    E. Kaucher. Über Eigenschaften und Anwendungsmöglichkeiten der erweiterten Intervallrechnung und des hyperbolischen Fastkörpers über ℝ. Comput. Suppl., 1:81–94, 1977.Google Scholar
  7. [7]
    E. Kaucher. Interval Analysis in the Extended Interval Space ℝ. Comput. Suppl., 2:33–49, 1980.MathSciNetGoogle Scholar
  8. [8]
    C. Lemaréchal and J. Zowe. The Eclipsing Concept to Approximate a Multi-Valued Mapping. Optimization, 22(1):3–37, 1991.MathSciNetCrossRefGoogle Scholar
  9. [9]
    S. M. Markov. On the presentation of ranges of monotone functions using interval arithmetic. Interval Comput., 4(6):19–31, 1992.Google Scholar
  10. [10]
    L. S. Pontryagin. Linear differential games. II. Soy. Math., Dokl., 8(4):910–912, 1967.Google Scholar
  11. [11]
    R. T. Rockafellar. Convex Analysis. Princeton Mathematical Series 28. Princeton University Press, Princeton, New Yersey, 1970.zbMATHGoogle Scholar
  12. [12]
    A. M. Rubinov and I. S. Akhundov. Difference of compact sets in the sense of Demyanov and its application to non-smooth analysis. Optimization, 23(3):179–188, 1992.MathSciNetCrossRefGoogle Scholar
  13. [13]
    D. B. Silin. On Set-Valued Differentiation and Integration. Set-Valued Anal., 5(2):107–146, 1997.MathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Robert Baier
    • 1
  • Elza M. Farkhi
    • 2
  1. 1.Chair of Applied MathematicsUniversity of BayreuthBayreuthGermany
  2. 2.School of Mathematical Sciences Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael

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