[Paper Review] 60 years of cyclic monotonicity: a survey

Luigi De Pascale, Anna Kausamo+1 more|arXiv (Cornell University)|Aug 15, 2023

Geometry and complex manifoldsMathematics3 citations

TL;DR

This survey provides a comprehensive overview of cyclic monotonicity over the past 60 years, emphasizing its foundational role in convex analysis and optimal transport theory. It establishes that c-cyclic monotonicity characterizes optimality in the Monge-Kantorovich transport problem, with Rockafellar's theorem proving that any maximal cyclically monotone set arises as the subdifferential of a convex function, thereby linking geometric optimality to convex duality.

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.

Motivation & Objective

• To provide a didactic synthesis of the state of the art in cyclic monotonicity and related concepts.
• To clarify the connection between cyclic monotonicity and optimality in the Monge-Kantorovich optimal transport problem.
• To present the historical development and key generalizations of cyclic monotonicity, including c-cyclic monotonicity.
• To establish the equivalence between c-cyclic monotonicity and optimality of transport plans in both L^1 and L^∞ settings.

Proposed method

• Defining cyclic monotonicity via the inequality ∑⟨x_i - x_{σ(i)}, y_i⟩ ≥ 0 for all permutations σ and all finite subsets of a set Γ ⊂ X×Y.
• Proving that subdifferentials of convex functions are cyclically monotone, and conversely, that any cyclically monotone set is contained in the subdifferential of some convex function (Rockafellar's theorem).
• Introducing c-cyclic monotonicity for general cost functions c, generalizing the quadratic cost case.
• Using discrete approximations of measures to extend results from rational to real coefficients, leveraging the density of rational solutions in the kernel of integer matrices.
• Applying permutation-based comparisons of transport costs to prove optimality via contradiction: if a plan is not c-cyclically monotone, a better plan exists.
• Extending results to multi-marginal optimal transport problems by constructing equivalent discrete measures with same marginals and comparing their total costs.

Experimental results

Research questions

• RQ1 How does cyclic monotonicity characterize the subdifferentials of convex functions, and what is the significance of maximality in this context?
• RQ2 What is the precise relationship between c-cyclic monotonicity and optimality in the Monge-Kantorovich transport problem?
• RQ3 How can the concept of cyclic monotonicity be generalized beyond the quadratic cost to arbitrary cost functions c?
• RQ4 In multi-marginal transport, how does c-cyclic monotonicity ensure optimality of a transport plan?
• RQ5 Can results for discrete measures with rational coefficients be extended to general real-valued measures?

Key findings

• Rockafellar's theorem establishes that every maximal cyclically monotone set is the subdifferential of a convex function, providing a complete characterization.
• A set Γ ⊂ X×Y is cyclically monotone if and only if ∑‖x_i - y_i‖² ≤ ∑‖x_i - y_{σ(i)}‖² for all permutations σ, linking the concept to the quadratic cost.
• c-Cyclic monotonicity is equivalent to optimality in the Kantorovich formulation of the optimal transport problem for both L^1 and L^∞ costs.
• For any finitely optimal transport plan γ, the support of γ must be c-cyclically monotone, as otherwise a better plan could be constructed via permutation.
• The proof technique relies on approximating general measures with rational-coefficient discrete measures, using the fact that rational solutions are dense in the kernel of integer matrices.
• The results hold in general topological vector spaces and extend to multi-marginal transport with N ≥ 2 marginals.