Cyclical monotonicity

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

Definition

Let $\langle\cdot,\cdot\rangle$ denote the inner product on an inner product space $X$ and let $U$ be a nonempty subset of $X$. A correspondence $f: U \rightrightarrows X$ is called ''cyclically monotone'' if for every set of points $x_1,\dots,x_ \in U$ with $x_=x_1$ it holds that $\sum_^m \langle x_,f(x_)-f(x_k)\rangle\geq 0.$

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 $U$ is convex and $f: U \rightrightarrows \mathbb^n$ is a correspondence with nonempty values. Then if $f$ is cyclically monotone, there exists an upper semicontinuous convex function $F:U\to \mathbb$ such that $f(x)\subset \partial F(x)$ for every $x\in U$, where $\partial F(x)$ denotes the subgradient of $F$ at $x$.http://www.its.caltech.edu/~kcborder/Courses/Notes/CyclicalMonotonicity.pdf

References

{{reflist

Mathematical terminology
Mathematical concepts