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60 years of cyclic monotonicity: a survey

L. De Pascale, A. Kausamo, K. Wyczesany

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.

60 years of cyclic monotonicity: a survey

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.
Paper Structure (22 sections, 22 theorems, 129 equations)

This paper contains 22 sections, 22 theorems, 129 equations.

Table of Contents

  1. Aims and scopes
  2. Cyclic monotonicity, potentials and generalizations
  3. Monotonicity and cyclic monotonicity
  4. Extensions of cyclic monotonicity
  5. $c$-cyclic monotonicity
  6. $c$-path-boundedness
  7. Adding more variables
  8. Classical optimal transport
  9. Optimality implies $c$-cyclic monotonicity
  10. When $c$-cyclic monotonicity implies optimality?
  11. Finitely-supported measures
  12. Purely atomic measures
  13. General measures
  14. Existence of plans supported on a $c$-subgradient
  15. More general transport problems
  16. ...and 7 more sections

Key Result

Theorem 2

Let $\, \Gamma \subset X \times Y$ be a cyclically monotone set. Then there exists a convex function $\varphi: X \to \mathbb R$ such that $\Gamma \subset \partial \varphi$.

Theorems & Definitions (53)

  • Definition 1
  • Theorem 2: rockafellar1966characterization, see also rockafellar1997convexrockafellar2015convex
  • Remark 3
  • Definition 4
  • Definition 5
  • Remark 6
  • Theorem 7: rochet, ruschendorf
  • proof
  • Definition 8
  • Definition 9
  • ...and 43 more