Cyclical monotonicity

Let $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ denote the inner product on an inner product space $X$ and let $U$ be a nonempty subset of $X$. A correspondence $f:\; U\; \backslash rightrightarrows\; X$ is called cyclically monotone if for every set of points $x\_1,\backslash dots,x\_\{m+1\}\; \backslash in\; U$ with $x\_\{m+1\}=x\_1$ it holds that $\backslash sum\_\{k=1\}^m\; \backslash langle\; x\_\{k+1\},f(x\_\{k+1\})-f(x\_k)\backslash rangle\backslash geq\; 0.${{cite book|page=9 |title=Revealed Preference Theory |year=2016 |publisher= Cambridge University Press |first1=Christopher P. |last1=Chambers |first2=Federico |last2=Echenique}}

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 $f:\; U\; \backslash rightrightarrows\; \backslash mathbb\{R\}^n$ is a correspondence with nonempty values. Then if $f$ is cyclically monotone, there exists an upper semicontinuous convex function $F:U\backslash to\; \backslash mathbb\{R\}$ such that $f(x)\backslash subset\; \backslash partial\; F(x)$ for every $x\backslash in\; U$, where $\backslash partial\; F(x)$ denotes the subgradient of $F$ at $x$.{{Cite book|title=Convex Analysis|last=Rockafellar|first= R. Tyrrell|author-link=R. Tyrrell Rockafellar|isbn=9781400873173|publisher=Princeton University Press|year=1970|contribution=Theorem 24.8|page=238}}

• Absolutely and completely monotonic functions and sequences

{{reflist}}

Category:Types of functions

Category:Linear algebra

Category:Functional analysis

{{Mathanalysis-stub}}