Further Conjugation Results

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


In this chapter, we exhibit several deeper results on conjugation. We first discuss Moreau’s decomposition principle, whereby a vector is decomposed in terms of the proximity operator of a lower semicontinuous function and that of its conjugate. This powerful nonlinear principle extends the standard linear decomposition with respect to a closed linear subspace and its orthogonal complement. Basic results concerning the proximal average and positively homogeneous functions are also presented. Also discussed are the Moreau– Rockafellar theorem, which characterizes coercivity in terms of an interiority condition, and the Toland–Singer theorem, which provides an appealing formula for the conjugate of the difference.


Homogeneous Function Nonempty Closed Convex Subset Lower Semicontinuous Function Proximity Operator Closed Linear Subspace 
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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Mathematics Irving K. Barber SchoolUniversity of British ColumbiaKelownaCanada
  2. 2.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie - Paris 6ParisFrance

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