Cyclical monotonicity

In mathematics, cyclical monotonicity is a generalization of the notion of monotonicity to the case of vector-valued function.^[1][2]

Definition

Let ${\displaystyle \langle \cdot ,\cdot \rangle }$ denote the inner product on an inner product space ${\displaystyle X}$ and let ${\displaystyle U}$ be a nonempty subset of ${\displaystyle X}$. A correspondence ${\displaystyle f:U\rightrightarrows X}$ is called cyclically monotone if for every set of points ${\displaystyle x_{1},\dots ,x_{m+1}\in U}$ with ${\displaystyle x_{m+1}=x_{1}}$ it holds that ${\displaystyle \sum _{k=1}^{m}\langle x_{k+1},f(x_{k+1})-f(x_{k})\rangle \geq 0.}$^[3]

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.^[4] Suppose ${\displaystyle U}$ is convex and ${\displaystyle f:U\rightrightarrows \mathbb {R} ^{n}}$ is a correspondence with nonempty values. Then if ${\displaystyle f}$ is cyclically monotone, there exists an upper semicontinuous convex function ${\displaystyle F:U\to \mathbb {R} }$ such that ${\displaystyle f(x)\subset \partial F(x)}$ for every ${\displaystyle x\in U}$, where ${\displaystyle \partial F(x)}$ denotes the subgradient of ${\displaystyle F}$ at ${\displaystyle x}$.^[5]

See also

• Absolutely and completely monotonic functions and sequences

References