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mathematics, cyclical monotonicity is a generalization of the notion of
monotonicity to the case of
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
- For the case of scalar functions of one variable the definition above is equivalent to usual
convex functions are cyclically monotone.
- In fact, the
converse is true.
 Suppose is
convex and is a correspondence with nonempty values. Then if is cyclically monotone, there exists an
upper semicontinuous convex function such that for every , where denotes the
subgradient of at .