---
title: Cyclic Quasi-Monotonicity
url: https://www.emergentmind.com/topics/cyclic-quasi-monotonicity
type: topic
---

# Cyclic Quasi-Monotonicity

Searching arXiv for recent and foundational papers on cyclic quasi-monotonicity and related notions.
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Cyclic quasi-monotonicity is a quasi-convex analogue of cyclic monotonicity for possibly multivalued maps \(F:\mathbb{R}^n \rightrightarrows \mathbb{R}^n\). In its precise recent formulation, it replaces the classical additive cycle inequality by a sign condition on the minimum directional pairing along a cycle: an operator is cyclically quasi-monotone when it admits no finite cycle whose successive displacements are all strictly positively aligned with selected values of the map. This notion is designed to connect monotonicity-type geometry with quasi-convex sublevel sets and their normal cones, in analogy with Rockafellar’s theorem for cyclically monotone operators and convex subdifferentials [2507.09437]. The terminology is not yet uniform across the literature: several nearby theories instead use weak cyclic monotonicity, \(N\)-cyclic monotonicity, \(c\)-cyclical monotonicity, or infinite \(c\)-cyclical monotonicity for distinct relaxations and cost-adapted generalizations of cyclic monotonicity [2308.07682].

## 1. Terminological scope and conceptual position

The term *cyclic quasi-monotonicity* is most sharply defined in the 2025 work on a Rockafellar-type theorem for cyclically quasi-monotone maps. There, the object of study is a multi-map \(F\) whose graph need not satisfy the classical cyclic sum inequality, but which still forbids “ascending” cycles in a min-type sense. The target representation is not a convex subdifferential \(\partial \phi\), but the normal cone operator \(N_f\) generated by sublevel sets of a quasi-convex function \(f\) [2507.09437].

This usage should be distinguished from several adjacent notions. The survey “60 years of cyclic monotonicity” explicitly notes that it does not introduce or use the term cyclic quasi-monotonicity; instead it organizes the area around 2-monotonicity, finite \(N\)-cyclic monotonicity, full cyclic monotonicity, \(c\)-cyclical monotonicity, infinite \(c\)-cyclical monotonicity, and structural relaxations such as \(c\)-connectivity and \(c\)-path-boundedness [2308.07682]. A common source of confusion is therefore terminological rather than mathematical: “quasi” may refer to a min/max replacement of the classical sum inequality, to finite-order relaxations, or to cost-adapted OT variants.

Conceptually, cyclic quasi-monotonicity sits strictly on the quasi-convex side of the convex-analysis taxonomy. Classical cyclic monotonicity is additive and subdifferential-based; cyclic quasi-monotonicity is order-theoretic and sublevel-based. The passage from convexity to quasi-convexity changes both the inequality and the representing operator: sums over cycles are replaced by a minimum test, and subgradients are replaced by normal cones to sublevel sets [2507.09437].

## 2. Definition, equivalent formulations, and basic geometry

Let \(F:\mathbb{R}^n \rightrightarrows \mathbb{R}^n\) be a multi-map. An \(F\)-ascending path is a finite sequence \(x_0,\dots,x_N\) together with selections \(p_i \in F(x_i)\) such that
$$
\langle p_i, x_{i+1}-x_i\rangle > 0 \quad \text{for all } 0 \le i < N.
$$
A cycle is such a path with \(x_N=x_0\). The map \(F\) is cyclically quasi-monotone if it has no ascending cycle. Equivalently, for every finite cycle \(x_0,\dots,x_N=x_0\) and every selection \(p_i \in F(x_i)\),
$$
\min_{0 \le i < N} \langle p_i, x_{i+1}-x_i\rangle \le 0.
$$
This is the quasi-analogue of the classical cyclic monotonicity inequality
$$
\sum_{i=0}^{N-1}\langle p_i, x_{i+1}-x_i\rangle \le 0,
$$
with the sum replaced by a minimum [2507.09437].

The same paper introduces an order-theoretic reformulation. Define a strict relation \(x \prec_F y\) if there exists an \(F\)-ascending path from \(x\) to \(y\). Then define
$$
x \preceq_F y \quad \Longleftrightarrow \quad \{w: w \prec_F x\} \subseteq \{w: w \prec_F y\}.
$$
Cyclic quasi-monotonicity is equivalent to the irreflexivity of \(\prec_F\), namely the exclusion of \(x \prec_F x\). From this relation one defines
$$
C^F(x):=\{y \in \mathbb{R}^n : y \preceq_F x\}.
$$
These sets are closed and convex, and they supply the geometric substrate on which quasi-convex potentials are built [2507.09437].

The associated quasi-convex normal cone operator is defined as follows. For a quasi-convex function \(f:\mathbb{R}^n \to \mathbb{R}\cup\{+\infty\}\), let
$$
C_f(x):=\{w \in \mathbb{R}^n : f(w)\le f(x)\},
$$
and define
$$
N_f(x):=N_{C_f(x)}(x)=\{p \in \mathbb{R}^n : f(w)\le f(x) \Rightarrow \langle p,w-x\rangle \le 0\}.
$$
This operator is the quasi-convex replacement for the convex subdifferential. If \(f\) is lower semicontinuous, its sublevel sets are closed; if not, replacing them by closures leaves the normal cone unchanged [2507.09437].

## 3. Rockafellar-type representation by quasi-convex potentials

The central structural question is whether cyclic quasi-monotonicity characterizes inclusion in a quasi-convex normal cone operator, just as classical cyclic monotonicity characterizes inclusion in a convex subdifferential. The main positive result currently available is a regular, non-vanishing theorem: if \(F:\mathbb{R}^n \to \mathbb{R}^n\) is a \(C^1\) vector field, cyclically quasi-monotone, and non-vanishing, then there exists a lower semicontinuous quasi-convex function \(f:\mathbb{R}^n \to \mathbb{R}\) such that
$$
F(x)\in N_f(x)\qquad \forall x\in\mathbb{R}^n.
$$
The same paper proves a one-dimensional theorem for general multi-maps \(F:\mathbb{R}\rightrightarrows\mathbb{R}\): no continuity, differentiability, or non-vanishing assumption is required to obtain a lower semicontinuous quasi-convex \(f\) with \(F(x)\subset N_f(x)\) for all \(x\) [2507.09437].

The construction proceeds in two stages. First, one shows that
$$
F(x)\subset N_{C^F(x)}(x)
$$
for every \(x\), where \(C^F(x)=\{y:y\preceq_F x\}\). Second, one seeks a quasi-convex function \(f\) whose spatially indexed sublevel sets coincide with the family \(C^F(x)\). This requires the family \(C^F\) to be totally ordered by inclusion. Under the regularity and non-vanishing assumptions, the paper proves exactly this total ordering by combining boundary geometry, pairwise interior intersection, and a perturbation argument along \(C^1\) boundary arcs [2507.09437].

Once total ordering is established, a representation theorem for totally ordered families of closed convex sets yields an lsc quasi-convex potential \(f\), unique only up to strictly increasing reparametrization. The resulting inclusion
$$
F\subset N_f
$$
is the quasi-convex analogue of Rockafellar’s classical relation \(F\subset \partial \phi\). The converse direction is also established: if \(f\) is quasi-convex and \(F\subset N_f\), then \(F\) is cyclically quasi-monotone. Thus quasi-convex normal cone operators are always CQM, while the reverse implication is fully proved in dimension one and in the \(C^1\), non-vanishing case in arbitrary dimension [2507.09437].

This shift from subgradients to normal cones is substantive rather than cosmetic. Subgradients encode additive support inequalities for convex functions; quasi-convex normal cones encode order relations among sublevel sets. A plausible implication is that cyclic quasi-monotonicity is best viewed as a geometric theory of ordered feasible regions rather than as a weakened calculus of convex differentials.

## 4. Neighboring notions and non-equivalent generalizations

Several notions neighboring cyclic quasi-monotonicity appear in the contemporary literature, but they are not interchangeable. The distinctions are structural.

| Notion | Defining test | Principal role |
|---|---|---|
| CQM | No ascending cycle; min-type inequality | Quasi-convex normal inclusion |
| WCM | Extend every CM sequence by a new point | Differential inclusions |
| \(N\)-cyclic monotonicity | Cycle inequalities up to fixed order \(N\) | Finite-order monotonicity, Hamiltonians |
| \(c\)-CM / ICM | Sum-type or max-type permutation inequalities | Optimal transport optimality |

Weak cyclic monotonicity (WCM), introduced for differential inclusions, fixes a base point \(x_0\) and a base value \(v_0\in F(x_0)\), defines cyclic monotone sequences by the classical inequality
$$
(x_m-x_0,v_m)\ge \sum_{i=1}^m (x_i-x_{i-1},v_{i-1}),
$$
and requires that every such finite sequence can be extended by an arbitrary new point \(x_m\) with a suitable choice \(v_m\in F(x_m)\). WCM generalizes cyclic monotonicity and is stronger than weak monotonicity. It is sufficient, together with compactness and upper semicontinuity, for local existence of solutions to differential inclusions with nonconvex right-hand side, via construction of a cyclically monotone subinclusion \(G\subset F\) [1307.2072].

The difference between WCM and CQM is fundamental. WCM is an extension property for classical cyclic monotone sequences and remains anchored in the additive inequality of convex analysis. CQM instead forbids all-positive directional cycles and is designed to recover quasi-convex, not convex, potentials. The constant multifunction \(F(x)=A\), with \(A\) compact and non-singleton, is WCM because one may continue a cyclic monotone sequence by repeating the previous value; yet it is not cyclically monotone, since monotone maps are almost everywhere single-valued [1307.2072]. This example illustrates that WCM is not simply another name for CQM.

Finite-order \(N\)-cyclic monotonicity is another nearby but distinct notion. For a vector field \(u:\Omega\to\mathbb{R}^d\), \(u\) is \(N\)-cyclically monotone if
$$
\sum_{i=1}^N \langle u(x_i),x_i-x_{i+1}\rangle \ge 0
$$
for every cycle \(x_1,\dots,x_N,x_{N+1}=x_1\). More generally, the 2012 paper studies jointly \(N\)-monotone \((N-1)\)-tuples and associates to them concave-convex \(N\)-sub-antisymmetric Hamiltonians \(H\) satisfying
$$
(u_1(x),\dots,u_{N-1}(x))=\nabla_{2,\dots,N}H(x,\dots,x)
$$
for almost every \(x\) [1207.2408]. This is a finite-order weakening of full cyclic monotonicity, not a quasi-convex theory in the sense of normal cones to sublevel sets.

In optimal transport, \(c\)-cyclical monotonicity is a cost-dependent generalization:
$$
\sum_{i=1}^k c(x_1^i,\dots,x_N^i)\le \sum_{i=1}^k c(x_1^i,x_2^{\sigma_2(i)},\dots,x_N^{\sigma_N(i)}),
$$
while infinite \(c\)-cyclical monotonicity replaces the sum by a maximum. For continuous costs and compactly supported marginals, sum-type \(c\)-cyclical monotonicity is sufficient for optimality in multimarginal Kantorovich transport, and max-type infinite \(c\)-cyclical monotonicity is sufficient for the \(L^\infty\) problem [2212.08375]. The survey literature treats these as cost-adapted or max-type analogues of cyclic monotonicity, not as instances of the 2025 CQM notion [2308.07682].

## 5. Optimal transport, perturbative viewpoints, and economic interpretation

A direct application of cyclic quasi-monotonicity appears in revealed preference theory. Given observed bundles \(x_i\) and prices \(p_i\), define \(F(x_i)=-p_i\). The 2025 paper states that finite datasets satisfying the Generalized Axiom of Revealed Preference are equivalent to the CQM property of this finite-domain operator, and that the data can then be rationalized by a quasi-convex utility \(-f\), with \(F(x_i)\in N_f(x_i)\) [2507.09437]. In this context, CQM is a geometric encoding of preference consistency: observed prices determine outward normals to utility sublevel sets.

The same paper also connects CQM to \(L^\infty\) optimal transport with correlation cost \(c(x,y)=-x\cdot y\). For a cycle \((x_i,y_i)\), the \(\infty\)-cyclical monotonicity condition
$$
\max_i c(x_i,y_i)\le \max_i c(x_i,y_i^{+})
$$
implies
$$
\min_i x_i\cdot (y_{i+1}-y_i)\le 0,
$$
which is exactly the CQM inequality when the “momentum” variable \(y\) is regarded as the selected value of a map. This does not identify \(L^\infty\) OT with CQM, but it exhibits a precise algebraic bridge between max-type transport inequalities and the min-type cycle test of cyclic quasi-monotonicity [2507.09437].

A broader perturbative perspective is provided by recent work on monotonicity in classical and weak optimal transport. There, \(c\)-cyclical monotonicity is characterized as non-negativity of the linear transport cost on the radial cone of admissible perturbations of a transport plan:
$$
\int c\, d\eta \ge 0 \quad \text{for all } \eta \in R_\pi,
$$
and the weak OT analogue becomes a first-order condition involving the linearized cost
$$
\int \nabla_\rho C(x,\pi_x)(y)\, d\eta(x,y)\ge 0 \quad \text{for all } \eta \in R_\pi.
$$
The authors do not use the term cyclic quasi-monotonicity, but this cone-restricted non-negativity suggests a directional interpretation of “quasi” optimality: positivity is required only along feasible perturbation directions, and finite cycle tests generate the whole condition [2606.19516].

This suggests a unifying theme. Classical cyclic monotonicity is additive and dual to convex potentials; \(c\)-cyclical monotonicity is cost-adapted and dual to \(c\)-convex potentials; cyclic quasi-monotonicity is order-based and dual to quasi-convex sublevel geometry. The common backbone is that cyclic constraints on finite configurations certify global optimality or integrability properties.

## 6. Examples, limitations, and open directions

The current theory includes several explicit examples. The “hedgehog” field,
$$
F(x)=\operatorname{span}_+(x)\quad (x\neq 0), \qquad F(0)=\mathbb{R}^n,
$$
is equal to the normal cone operator of \(|\cdot|\), hence is CQM. The “single circle” field in \(\mathbb{R}^2\), given by \(F(x)=x\) on \(|x|=1\) and \(F(x)=0\) otherwise, satisfies \(F\subset N_f\) for multiple quasi-convex potentials, such as \(f(x)=|x|\) or a two-level indicator-type potential. Other constructions include the plateau-and-jump example, the stadium field, the three-quarters hedgehog, the half hedgehog, and a half-hedgehog–half-constant field [2507.09437].

These examples show that existence of a quasi-convex potential can persist even when the family \(C^F(x)\) is not totally ordered, but then the general construction via ordered convex families may fail. They also show that closures of multivalued graphs can enlarge normal cones on boundaries, sometimes forcing discontinuous lower semicontinuous potentials. In this sense, the regular non-vanishing theorem is a sufficiency result, not a full characterization of all higher-dimensional multi-maps [2507.09437].

An important neighboring example comes from weak cyclic monotonicity rather than CQM. If \(A\subset \mathbb{R}^n\) is compact and \(F(x)=A\) for all \(x\), then \(F\) is weakly cyclic monotone because one may choose \(v_m=v_{m-1}\) and preserve the classical cyclic monotone-sequence condition, yet \(F\) is not cyclically monotone when \(A\) is not a singleton [1307.2072]. This example is often useful for preventing overidentification of weak, cyclic, and quasi variants.

Several open problems remain explicit. For cyclic quasi-monotonicity in the sense of quasi-convex normal inclusion, the central unresolved directions are removing the non-vanishing hypothesis in higher dimensions, lowering regularity from \(C^1\) to continuity or weaker assumptions, understanding maximality and equality cases for the operator \(N_f\) containing \(F\), and classifying all quasi-convex potentials associated with a given CQM map [2507.09437]. More broadly, the survey literature indicates that any use of the term “cyclic quasi-monotonicity” should be interpreted carefully against the established vocabularies of finite-order cyclic monotonicity, weak cyclic monotonicity, and cost-based cyclical monotonicity [2308.07682].

The present state of the subject therefore has a clear structure. Cyclic quasi-monotonicity, in the strict recent sense, is the no-ascending-cycle condition that links vector fields and multi-maps to quasi-convex sublevel geometry. It is neither a synonym for weak cyclic monotonicity nor a synonym for \(c\)-cyclical monotonicity, though all three belong to the wider family of cyclic inequalities and cyclic extension principles. Its distinctive feature is the replacement of convex additivity by quasi-convex order, and its distinctive operator is the normal cone to a sublevel set rather than the convex subdifferential [2507.09437].

Source: https://www.emergentmind.com/topics/cyclic-quasi-monotonicity