Cyclical monotonicity

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

Definition

Let [math]\displaystyle{ \langle\cdot,\cdot\rangle }[/math] denote the inner product on an inner product space [math]\displaystyle{ X }[/math] and let [math]\displaystyle{ U }[/math] be a nonempty subset of [math]\displaystyle{ X }[/math]. A correspondence [math]\displaystyle{ f: U \rightrightarrows X }[/math] is called cyclically monotone if for every set of points [math]\displaystyle{ x_1,\dots,x_{m+1} \in U }[/math] with [math]\displaystyle{ x_{m+1}=x_1 }[/math] it holds that [math]\displaystyle{ \sum_{k=1}^m \langle x_{k+1},f(x_{k+1})-f(x_k)\rangle\geq 0. }[/math]

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

See also

• Absolutely and completely monotonic functions and sequences