User:Грихан/sandbox

Cyclical monotonicity is one possible generalization of the notion of monotonicity to the case of vector functions from $$\mathbb{R}^n$$to $$\mathbb{R}^n$$.

Definition
Let $$\langle\cdot,\cdot\rangle$$denote the standard dot product in $$\mathbb{R}^n$$. A function $$f: U\subset \mathbb{R}^n \to \mathbb{R}^n$$is called cyclically monotone if for every set of points $$x_1,\dots,x_{k+1} \in U

$$with $$x_{k+1}=x_1$$ it holds that $$\sum_{i=1}^k \langle x_{k+1},f(x_{k+1})-f(x_k)\rangle\geq 0 $$

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. Suppose $$U$$is convex and $$f: U \rightrightarrows \mathbb{R}^n$$is a correspondence with nonempty values. Then if $$f$$ is cyclically monotone, then there exists an upper semicontinuous convex function $$F:U\to \mathbb{R}$$such that $$f(x)\subset \partial F(x)$$for every $$x\in U$$