Stronger Notions of Monotonicity

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


This chapter collects basic results on various stronger notions of monotonicity (para, strict, uniform, strong, and cyclic) and their relationships to properties of convex functions. A fundamental result is Rockafellar’s characterization of maximally cyclically monotone operators as subdifferential operators and a corresponding uniqueness result for the underlying convex function.


  1. [7]
    E. Asplund, A monotone convergence theorem for sequences of nonlinear mappings, in Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 1–9.Google Scholar
  2. [39]
    H. H. Bauschke, J. M. Borwein, and X. Wang, Fitzpatrick functions and continuous linear monotone operators, SIAM J. Optim., 18 (2007), pp. 789–809.MathSciNetCrossRefGoogle Scholar

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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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