Cyclical monotonicity

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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 yields 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. 7. Germany: Springer Science+Business Media: 7–32. doi:10.1023/A:1008753021652.
  2. ^ Beiglböck, Mathias (May 2015). "Cyclical monotonicity and the ergodic theorem". Ergodic Theory and Dynamical Systems. 35 (3). Cambridge University Press: 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.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)[page needed]
  5. ^ http://www.its.caltech.edu/~kcborder/Courses/Notes/CyclicalMonotonicity.pdf