Abstract

The primary purpose of this note is to provide an instructional summary of the state of the art regarding cyclic monotonicity and related notions. We will also present how these notions are tied to optimality in the optimal transport (or Monge-Kantorovich) problem, even though cyclic monotonicity is simply a property of a set and can be studied on its own merit.

Author information

Contact details are reproduced from the original publication and may be historical.

Luigi De Pascale

Dip. di Matematica ed Informatica, Università degli Studi di Firenze, Firenze, Italy

luigi.depascale@unifi.it

Anna Kausamo

Dip. di Matematica ed Informatica, Università degli Studi di Firenze, Firenze, Italy

akausamo@gmail.com

Katarzyna Wyczesany

Dept. of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, U.S.A.

kwycz@cmu.edu

Keywords

  • Monge–Kantorovich problem
  • optimality condition
  • c-cyclical monotonicity
  • Kantorovich potentials
  • transport map

Mathematics Subject Classification

49J45, 49N15, 49K21, 49K27, 49Q22

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Suggested citation

L. De Pascale, A. Kausamo, K. Wyczesany. “60 Years of Cyclic Monotonicity: a Survey.” Journal of Convex Analysis 32 (2025), No. 2, 399–430. https://doi.org/10.68381/jca32020

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