Cyclical monotonicity

From Wikipedia
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 denote the inner product on an inner product space and let be a nonempty subset of . A correspondence is called cyclically monotone if for every set of points with it holds that [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 is convex and is a correspondence with nonempty values. Then if is cyclically monotone, then there exists an upper semicontinuous convex function such that for every , where denotes the subgradient of at . [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. ^ http://www.its.caltech.edu/~kcborder/Courses/Notes/CyclicalMonotonicity.pdf