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Conjugation

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

Abstract

Functional transforms make it possible to investigate problems from a different perspective and sometimes simplify their investigation. In convex analysis, the most suitable notion of a transform is the Legendre transform, which maps a function to its Fenchel conjugate. This transform is studied in detail in this chapter. In particular, it is shown that the conjugate of an infimal convolution is the sum of the conjugates. The key result of this chapter is the Fenchel–Moreau theorem, which states that the proper convex lower semicontinuous functions are precisely those functions that coincide with their biconjugates.

References

  1. [317]
    R. T. Rockafellar, Convex integral functionals and duality, in Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, ed., Academic Press, New York, 1971, pp. 215–236.CrossRefGoogle Scholar
  2. [318]
    R. T. Rockafellar, Conjugate Duality and Optimization, SIAM, Philadelphia, PA, 1974.CrossRefGoogle Scholar

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© Springer International Publishing AG 2017

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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

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