Advertisement

Convex Cones and Generalized Interiors

  • Heinz H. Bauschke
  • Patrick L. Combettes
Chapter
Part of the CMS Books in Mathematics book series (CMSBM)

Abstract

The notion of a convex cone, which lies between that of a linear subspace and that of a convex set, is the main topic of this chapter. It has been very fruitful in many branches of nonlinear analysis. For instance, closed convex cones provide decompositions analogous to the well-known orthogonal decomposition based on closed linear subspaces. They also arise naturally in convex analysis in the local study of a convex set via the tangent cone and the normal cone operators, and they are central in the analysis of various extensions of the notion of an interior that will be required in later chapters.

References

  1. [69]
    J. M. Borwein and A. S. Lewis, Partially finite convex programming I – Quasi relative interiors and duality theory, Math. Programming Ser. B, 57 (1992), pp. 15–48.MathSciNetCrossRefGoogle Scholar
  2. [205]
    R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1990.zbMATHGoogle Scholar
  3. [313]
    R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.CrossRefGoogle Scholar
  4. [329]
    S. Simons, Minimax and Monotonicity, vol. 1693 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1998.CrossRefGoogle Scholar
  5. [365]
    E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets, in Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, ed., Academic Press, New York, 1971, pp. 237–341.Google Scholar

Copyright information

© Springer International Publishing AG 2017

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 2.5 International License (http://creativecommons.org/licenses/by-nc/2.5/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Authors and Affiliations

  • Heinz H. Bauschke
    • 1
  • Patrick L. Combettes
    • 2
  1. 1.Department of MathematicsUniversity of British ColumbiaKelownaCanada
  2. 2.Department of MathematicsNorth Carolina State UniversityRaleighUSA

Personalised recommendations