Cyclical monotonicity

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

## Contents

* 1 Definition
* 2 Properties
* 3 See also
* 4 References

## Definition

Let ⟨⋅,⋅⟩ denote the inner product on an inner product space X and let U be a nonempty subset of X. A correspondence f:U⇉X is called cyclically monotone if for every set of points x1,…,xm+1∈U with xm+1=x1 it holds that ∑k=1m⟨xk+1,f(xk+1)−f(xk)⟩≥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 U is convex and f:U⇉ℝn is a correspondence with nonempty values. Then if f is cyclically monotone, there exists an upper semicontinuous convex function F:U->ℝ such that f(x)⊂∂F(x) for every x∈U, where ∂F(x) denotes the subgradient of F at x.[5]

## See also

* Absolutely and completely monotonic functions and sequences

## References

1. ↑ Levin, Vladimir (1 March 1999). "Abstract Cyclical Monotonicity and Monge Solutions for the General Monge–Kantorovich Problem" (in en). 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" (in en). 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. [page needed]
5. ↑ http://www.its.caltech.edu/~kcborder/Courses/Notes/CyclicalMonotonicity.pdf [bare URL PDF]

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Cyclical monotonicity. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Cyclical_monotonicity&oldid=237501"