mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, cyclical monotonicity is a generalization of the notion of monotonicity 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, ofte ...

X

and let

U

be a

nonempty In mathematics, the empty set or void 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, whi ...

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.

Gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...

s of

convex function In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is conve ...

s are cyclically monotone. In fact, the converse is true. Suppose

U

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 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 In mathematics, the subderivative (or subgradient) generalizes the derivative to convex functions which are not necessarily differentiable. The set of subderivatives at a point is called the subdifferential at that point. Subderivatives arise in c ...

F

x

References

Types of functions Linear algebra Functional analysis {{Mathanalysis-stub

Definition

Properties

See also

References