Abstract
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.
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References
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.
H. H. Bauschke, J. M. Borwein, and X. Wang, Fitzpatrick functions and continuous linear monotone operators, SIAM J. Optim., 18 (2007), pp. 789–809.
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Bauschke, H.H., Combettes, P.L. (2017). Stronger Notions of Monotonicity. In: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-48311-5_22
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DOI: https://doi.org/10.1007/978-3-319-48311-5_22
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