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.
In fact, the
converse is true.[4] 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 .[5]