Cyclical Monotonicity in Optimal Transport

• Cyclical monotonicity is a mathematical property that refines classical monotonicity by ensuring no cyclic reordering of paired elements reduces total cost in convex analysis and transport problems.
• It provides a geometric and variational framework that is pivotal for establishing necessary and sufficient optimality conditions in both classical and weak optimal transport settings.
• Its generalizations extend to multi-marginal problems, port-Hamiltonian systems, and differential inclusions, demonstrating broad applicability in diverse optimization and control contexts.

Cyclical monotonicity is a central concept in convex analysis, optimal transport theory, and the study of monotone operators, providing a geometric and variational characterization of optimal transport plans and subdifferentials of convex functions. Its generalizations underpin the structure of optimizers in classical and weak transport problems, multi-marginal settings, port-Hamiltonian systems, and even differential inclusions with non-convex right-hand sides.

1. Definition and Basic Properties

Cyclical monotonicity refines classical monotonicity of set-valued maps and relations. For a pair of spaces (typically $X$ and $Y$ vector spaces or Polish spaces) and a function $c\colon X \times Y \to \mathbb{R} \cup \{+\infty\}$, a set $\Gamma \subset X \times Y$ is $c$-cyclically monotone if for any finite collection $\{(x_i, y_i)\}_{i=1}^n \subset \Gamma$ and any permutation $\sigma$ of $\{1,\ldots,n\}$,

$$\sum_{i=1}^n c(x_i, y_i) \leq \sum_{i=1}^n c(x_i, y_{\sigma(i)}).$$

In the classical monotonicity case ($c(x, y) = -\langle x, y \rangle$), this reduces to the usual monotonicity inequality. The set is maximally $Y$0-cyclically monotone if it cannot be strictly enlarged without losing this property. The concept extends naturally to multi-marginal settings and infinite-dimensional linear programs by a suitable adaptation of the cycle concept, e.g., by permutations of coordinates in the product $Y$1 (Pascale et al., 2023, Beiglböck et al., 2014, Pascale et al., 2022).

Key properties:

• In convex analysis, the graph of the subdifferential $Y$2 of a proper lower semicontinuous convex function $Y$3 is maximally cyclically monotone (Pascale et al., 2023, Camlibel et al., 2022).
• In optimal transport, cyclically monotone supports of transport plans characterize optimality for broad classes of cost functions (Veraguas et al., 2018, Ehrlacher et al., 17 Jun 2026, Chi et al., 2021).

2. Role in Optimal Transport Theory

Cyclical monotonicity provides both necessary and sufficient conditions for optimality in the Monge–Kantorovich transport problem: $Y$4 where $Y$5 ranges over couplings of given marginals $Y$6.

• Necessity: Any optimal coupling $Y$7 is $Y$8-cyclically monotone, regardless of whether the cost $Y$9 is merely Borel measurable and nonnegative. This follows from a cycle perturbation argument: if the inequality fails on any finite cycle, a small mass rearrangement along the cycle reduces the cost, contradicting optimality (Ehrlacher et al., 17 Jun 2026, Pascale et al., 2023).
• Sufficiency: Under additional regularity (e.g., lower semicontinuity, compactness of support, or bounded growth), any $c\colon X \times Y \to \mathbb{R} \cup \{+\infty\}$0-cyclically monotone plan of finite cost is optimal. Sufficiency fails if no growth bound is imposed or if the cost can take $c\colon X \times Y \to \mathbb{R} \cup \{+\infty\}$1 values without being path-bounded (Pascale et al., 2023, Griessler, 2016, Pascale et al., 2022, Beiglböck et al., 2014).

This principle unifies the variational, geometric, and dual (potential-theoretic) perspectives and underlies efficient solution methods for finite, infinite, classical, and multi-marginal transport (Beiglböck et al., 2014, Pascale et al., 2022, Chi et al., 2021).

3. Duality and Convex Potentials

Rockafellar’s theorem establishes that maximal cyclically monotone sets are precisely subdifferential graphs of convex functions. In the classical setting ($c\colon X \times Y \to \mathbb{R} \cup \{+\infty\}$2), a set $c\colon X \times Y \to \mathbb{R} \cup \{+\infty\}$3 is maximally cyclically monotone if and only if there exists a proper l.s.c. convex function $c\colon X \times Y \to \mathbb{R} \cup \{+\infty\}$4 such that $c\colon X \times Y \to \mathbb{R} \cup \{+\infty\}$5. The corresponding variational formula for $c\colon X \times Y \to \mathbb{R} \cup \{+\infty\}$6 is built from chain sums along cycles in $c\colon X \times Y \to \mathbb{R} \cup \{+\infty\}$7 (Camlibel et al., 2022, Pascale et al., 2023).

In optimal transport with general cost $c\colon X \times Y \to \mathbb{R} \cup \{+\infty\}$8, the analogous notion is $c\colon X \times Y \to \mathbb{R} \cup \{+\infty\}$9-convexity. For a $\Gamma \subset X \times Y$0-cyclically monotone set $\Gamma \subset X \times Y$1, there exists a $\Gamma \subset X \times Y$2-convex function $\Gamma \subset X \times Y$3 (satisfying Fenchel-type inequalities) such that $\Gamma \subset X \times Y$4. If $\Gamma \subset X \times Y$5 is real-valued and continuous, maximality holds, i.e., $\Gamma \subset X \times Y$6 (Pascale et al., 2023).

In weak optimal transport, duality involves a functional $\Gamma \subset X \times Y$7 (convex in $\Gamma \subset X \times Y$8) and the criterion of $\Gamma \subset X \times Y$9-monotonicity, tightly coupled to complementary slackness in the dual (Veraguas et al., 2018, Ehrlacher et al., 17 Jun 2026). The dual potential $c$0 is minimized at the disintegration $c$1, and the optimality conditions are localized via cyclical monotonicity (Veraguas et al., 2018).

4. Generalizations: Weak, Multi-Marginal, and Constrained Settings

The cyclical monotonicity principle extends to several advanced settings:

• Weak Transport: The cost is $c$2 for $c$3 and cyclical monotonicity is replaced by a "no-improvement" inequality involving alternative conditional distributions, matching the structure of the weak cost functional (Veraguas et al., 2018, Ehrlacher et al., 17 Jun 2026).
• Multi-Marginal Transport: Cyclical monotonicity is generalized to product spaces $c$4 with cost $c$5. The corresponding criterion involves simultaneous permutations of components and is necessary and sufficient for optimality under mild regularity (Griessler, 2016, Pascale et al., 2022).
• Capacity Constraints: In constrained OT, $c$6-cyclical monotonicity is further generalized to $c$7-capacity monotonicity. Here admissible perturbations are constrained to "competitor" measures respecting both marginals and capacity-induced costs. The constrained optimizer is $c$8-capacity monotone, nesting the unconstrained theory as a special case (Chen, 28 Aug 2025).

A unifying notion in all these settings is finitistic optimality, which requires cyclic monotonicity for all finite configurations compatible with the constraints of the problem (Beiglböck et al., 2014).

5. Interactions with Monotone Operator Theory and Convex Analysis

Cyclical monotonicity lies at the interface of convex analysis and the theory of monotone operators:

• Maximal Cyclically Monotone Relations: In finite dimensions, Rockafellar’s theorem implies every maximal cyclically monotone relation is the subdifferential of a convex function, and this characterization extends to port-Hamiltonian systems, gradient flows, and electrical networks. The construction of the generating convex potential uses a sup formula over chains in the relation (Camlibel et al., 2022).
• Multi-Conjugate and Multi-Marginal Theory: In the multi-marginal context, cyclically monotone contact sets correspond to multi-conjugate convex functions. On the real line, these properties extend the Fenchel–Moreau involution (i.e., $c$9). In higher dimensions, additional smoothness (essential smoothness) is required for involutivity and maximality; otherwise, contact sets can fail to be maximal cyclically monotone (Lim, 2022).
• Weak Cyclic Monotonicity: Further weakening, as in differential inclusions, a map is weakly cyclically monotone if every finite cyclically monotone chain can be extended by at least one more point while maintaining monotonicity. This property, strictly weaker than classical cyclic monotonicity but stronger than one-sided monotonicity, allows for existence results in otherwise non-convex settings (Farkhi, 2013).

6. Algorithmic, Structural, and Applied Consequences

Cyclical monotonicity has algorithmic and structural significance:

• Structure of Optimal Plans: In $\{(x_i, y_i)\}_{i=1}^n \subset \Gamma$0-dimensional OT, the optimal coupling is supported on a cyclically monotone set, often non-crossing in the one-dimensional case, or a permutation in discrete cases when additional structure (e.g., conditional negative semi-definiteness) is present (Vayer, 18 Feb 2026).
• Port-Hamiltonian and Equilibrium-Independent Passive Systems: In incremental passivity analysis and interconnected physical systems, maximal cyclically monotone relations and their composition via convex potentials provide the underlying variational structure (Camlibel et al., 2022).
• Practical Optimization: Enforcing or penalizing violations of $\{(x_i, y_i)\}_{i=1}^n \subset \Gamma$1-cyclical monotonicity leads to more tractable and meaningful regularized transport problems, e.g., in Wasserstein barycenter computation (Chi et al., 2021).

A summary of core facts is given in the following table.

Setting	Cyclical Monotonicity Role	Sufficient for Optimality?
Classical OT (two marg.)	Necessary and sufficient (with regularity)	Yes, under lower-semicontinuity or boundedness (Pascale et al., 2023, Ehrlacher et al., 17 Jun 2026)
Multi-marginal OT	Necessary and sufficient (with regularity)	Yes, for continuous costs and growth bounds (Griessler, 2016, Pascale et al., 2022)
Weak OT	$\{(x_i, y_i)\}_{i=1}^n \subset \Gamma$2-monotonicity as first-order criterion	Yes, under convexity and regularity (Veraguas et al., 2018, Ehrlacher et al., 17 Jun 2026)
Port-Hamiltonian	Characterizes subdifferential structure	Yes, links to incremental passivity (Camlibel et al., 2022)
Capacity constraints	$\{(x_i, y_i)\}_{i=1}^n \subset \Gamma$3-capacity monotonicity generalizes $\{(x_i, y_i)\}_{i=1}^n \subset \Gamma$4-cyclic	Yes, in the presence of the capacity constraint (Chen, 28 Aug 2025)
Differential inclusions	Weak cyclic monotonicity gives existence	Sufficient for existence, weaker than classical monotonicity (Farkhi, 2013)

7. Open Problems and Further Developments

Key open questions in the theory include:

• Sufficiency of cyclical monotonicity without continuity assumptions on the cost in multi-marginal and infinite-dimensional settings (Pascale et al., 2023, Pascale et al., 2022).
• Characterization and structure of maximal $\{(x_i, y_i)\}_{i=1}^n \subset \Gamma$5-cyclically monotone sets in infinite-dimensional or non-separable spaces (Pascale et al., 2023).
• Quantitative and stability estimates for monotonicity, and connections to regularity in PDEs and control theory (Pascale et al., 2023).
• Complete duality theory in settings with partial or irregular marginal data and generalized moment problems (Beiglböck et al., 2014).

Cyclical monotonicity remains a unifying concept illuminating the intersection of convex geometry, variational analysis, and optimization. In both foundational theoretical and applied computational regimes, it identifies the geometric locus of optimality and underlies the emergence of dual potentials and monotone relations characteristic of optimal transport and convex variational systems.