Multisets and Multiset Mappings

  • Adam B. Levy
Part of the SpringerBriefs in Optimization book series (BRIEFSOPTI)


Multisets are collections of elements which may appear with multiplicity. We review the multiplicity function and cardinality, and we introduce the notion of filter to extract the distinct elements in a multiset. We define a notion of outer limit for sequences of multisets, and we define and develop “multiset mappings” that take multisets to multisets. We introduce a generalized continuity property for multiset mappings (called calmness) that relies on choices of pre-distance functions, and we investigate how this property generalizes a similar property for set-valued mappings. We define a generalized derivative for multiset mappings and use it to characterize calmness.


  1. 13.
    Knuth, D.E.: Seminumerical Algorithms. The Art of Computer Programming, vol. 2, 3rd edn. Addison-Wesley, Boston (1998)Google Scholar
  2. 20.
    Robinson, S.M.: Generalized equations and their solutions. I. Basic theory. In: Point-to-Set Maps and Mathematical Programming. Mathematical Programming Studies, vol. 10, pp. 128–141. Springer, Berlin (1979)Google Scholar
  3. 21.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefGoogle Scholar

Copyright information

© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Adam B. Levy
    • 1
  1. 1.Bowdoin CollegeBrunswickUSA

Personalised recommendations