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]
In mathematics, cyclical monotonicity is a generalization of the notion of monotonicity to the case of vector-valued function. [1] [2]
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]
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, there exists an upper semicontinuous convex function such that for every , where denotes the subgradient of at . [5]
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