Английская Википедия:Cyclical monotonicity

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

Definition

Let <math>\langle\cdot,\cdot\rangle</math> denote the inner product on an inner product space <math>X</math> and let <math>U</math> be a nonempty subset of <math>X</math>. A correspondence <math>f: U \rightrightarrows X</math> is called cyclically monotone if for every set of points <math>x_1,\dots,x_{m+1} \in U</math> with <math>x_{m+1}=x_1</math> it holds that <math>\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>U</math> is convex and <math>f: U \rightrightarrows \mathbb{R}^n</math> is a correspondence with nonempty values. Then if <math>f</math> is cyclically monotone, there exists an upper semicontinuous convex function <math>F:U\to \mathbb{R}</math> such that <math>f(x)\subset \partial F(x)</math> for every <math>x\in U</math>, where <math>\partial F(x)</math> denotes the subgradient of <math>F</math> at <math>x</math>.

See also

• Absolutely and completely monotonic functions and sequences