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 $x_{1},\dots ,x_{m+1}\in U$ with $x_{m+1}=x_{1}$ it holds that $\sum _{k=1}^{m}\langle x_{k+1},f(x_{k+1})-f(x_{k})\rangle \geq 0.$

Properties

• For the case of scalar functions of one variable the definition above is equivalent to usual monotonicity.
• Gradients of convex functions are cyclically monotone.
• In fact, the converse is true. Suppose $U$ is convex and $f:U\rightrightarrows \mathbb {R} ^{n}$ is a correspondence with nonempty values. Then if $f$ is cyclically monotone, there exists an upper semicontinuous convex function $F:U\to \mathbb {R}$ such that $f(x)\subset \partial F(x)$ for every $x\in U$, where $\partial F(x)$ denotes the subgradient of $F$ at $x$.