Cyclical monotonicity
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In mathematics, cyclical monotonicity is a generalization of the notion of
monotonicity In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
to the case of
vector-valued function A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
.


Definition

Let \langle\cdot,\cdot\rangle denote the inner product on an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
X and let U be a
nonempty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
subset of X. A correspondence f: U \rightrightarrows X is called ''cyclically monotone'' if for every set of points x_1,\dots,x_ \in U with x_=x_1 it holds that \sum_^m \langle x_,f(x_)-f(x_k)\rangle\geq 0.


Properties

* For the case of scalar functions of one variable the definition above is equivalent to usual
monotonicity In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
. *
Gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
s of convex functions are cyclically monotone. * In fact, the
converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical c ...
is true. Suppose U is
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
and f: U \rightrightarrows \mathbb^n is a correspondence with nonempty values. Then if f is cyclically monotone, there exists an
upper semicontinuous In mathematical analysis, semicontinuity (or semi-continuity) is a property of Extended real number, extended real-valued Function (mathematics), functions that is weaker than Continuous function, continuity. An extended real-valued function f is ...
convex function F:U\to \mathbb such that f(x)\subset \partial F(x) for every x\in U, where \partial F(x) denotes the subgradient of F at x.http://www.its.caltech.edu/~kcborder/Courses/Notes/CyclicalMonotonicity.pdf


References

{{reflist Mathematical terminology Mathematical concepts