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

Properties

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

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References

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.
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.
Chambers, Christopher P.; Echenique, Federico (2016). Revealed Preference Theory. Cambridge University Press. p. 9.
Rockafellar, R. Tyrrell, 1935- (2015-04-29). Convex analysis. Princeton, N.J. ISBN 9781400873173. OCLC 905969889.CS1 maint: multiple names: authors list (link)
http://www.its.caltech.edu/~kcborder/Courses/Notes/CyclicalMonotonicity.pdf

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[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.

[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)

[5] ttp://www.its.caltech.edu/~kcborder/Courses/Notes/CyclicalMonotonicity.pdf