In mathematics, cyclical monotonicity is a generalization of the notion of monotonicity to the case of vector-valued function. 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 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. 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 . References Mathematical terminology Mathematical concepts