# Cyclical monotonicity

https://en.wikipedia.org/wiki/Cyclical_monotonicity

In mathematics, cyclical monotonicity is a generalization of the notion of monotonicity to the case of vector-valued function. [1] [2]

## Definition

Let ${\displaystyle \langle \cdot ,\cdot \rangle }$ denote the inner product on an inner product space ${\displaystyle X}$ and let ${\displaystyle U}$ be a nonempty subset of ${\displaystyle X}$. A correspondence ${\displaystyle f:U\rightrightarrows X}$ is called cyclically monotone if for every set of points ${\displaystyle x_{1},\dots ,x_{m+1}\in U}$ with ${\displaystyle x_{m+1}=x_{1}}$ it holds that ${\displaystyle \sum _{k=1}^{m}\langle x_{k+1},f(x_{k+1})-f(x_{k})\rangle \geq 0.}$ [3]

## Properties

• For the case of scalar functions of one variable the definition above is equivalent to usual monotonicity.
• Gradients of convex functions are cyclically monotone.
• In fact, the converse is true. [4] Suppose ${\displaystyle U}$ is convex and ${\displaystyle f:U\rightrightarrows \mathbb {R} ^{n}}$is a correspondence with nonempty values. Then if ${\displaystyle f}$ is cyclically monotone, then there exists an upper semicontinuous convex function ${\displaystyle F:U\to \mathbb {R} }$ such that ${\displaystyle f(x)\subset \partial F(x)}$ for every ${\displaystyle x\in U}$, where ${\displaystyle \partial F(x)}$ denotes the subgradient of ${\displaystyle F}$ at ${\displaystyle x}$. [5]

## References

1. ^ Levin, Vladimir (1 March 1999). "Abstract Cyclical Monotonicity and Monge Solutions for the General Monge–Kantorovich Problem". Set-Valued Analysis. Germany: Springer Science+Business Media. 7: 7–32. doi: 10.1023/A:1008753021652. S2CID  115300375.
2. ^ Beiglböck, Mathias (May 2015). "Cyclical monotonicity and the ergodic theorem". Ergodic Theory and Dynamical Systems. Cambridge University Press. 35 (3): 710–713. doi: 10.1017/etds.2013.75.
3. ^ Chambers, Christopher P.; Echenique, Federico (2016). Revealed Preference Theory. Cambridge University Press. p. 9.
4. ^ Rockafellar, R. Tyrrell, 1935- (2015-04-29). Convex analysis. Princeton, N.J. ISBN  9781400873173. OCLC  905969889.CS1 maint: multiple names: authors list ( link)[ page needed]
5. ^