Set-valued function

Last updated October 31, 2024

A set-valued function, also called a correspondence or set-valued relation , is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set.^[1]^[2] Set-valued functions are used in a variety of mathematical fields, including optimization, control theory and game theory.

Distinction from multivalued functions

Although other authors may distinguish them differently (or not at all), Wriggers and Panatiotopoulos (2014) distinguish multivalued functions from set-valued functions (which they called set-valued relations) by the fact that multivalued functions only take multiple values at finitely (or denumerably) many points, and otherwise behave like a function.^[2] Geometrically, this means that the graph of a multivalued function is necessarily a line of zero area that doesn't loop, while the graph of a set-valued relation may contain solid filled areas or loops.^[2]

Alternatively, a multivalued function is a set-valued function $f$ that has a further continuity property, namely that the choice of an element in the set $f(x)$ defines a corresponding element in each set $f(y)$ for $y$ close to $x$ , and thus defines locally an ordinary function.

Example

The argmax of a function is in general, multivalued. For example, $\operatorname {argmax} _{x\in \mathbb {R} }\cos(x)=\{2\pi k\mid k\in \mathbb {Z} \}$ .

Set-valued analysis

Set-valued analysis is the study of sets in the spirit of mathematical analysis and general topology.

Instead of considering collections of only points, set-valued analysis considers collections of sets. If a collection of sets is endowed with a topology, or inherits an appropriate topology from an underlying topological space, then the convergence of sets can be studied.

Much of set-valued analysis arose through the study of mathematical economics and optimal control, partly as a generalization of convex analysis; the term "variational analysis" is used by authors such as R. Tyrrell Rockafellar and Roger J-B Wets, Jonathan Borwein and Adrian Lewis, and Boris Mordukhovich. In optimization theory, the convergence of approximating subdifferentials to a subdifferential is important in understanding necessary or sufficient conditions for any minimizing point.

There exist set-valued extensions of the following concepts from point-valued analysis: continuity, differentiation, integration,^[4] implicit function theorem, contraction mappings, measure theory, fixed-point theorems,^[5] optimization, and topological degree theory. In particular, equations are generalized to inclusions, while differential equations are generalized to differential inclusions.

One can distinguish multiple concepts generalizing continuity, such as the closed graph property and upper and lower hemicontinuity ^{[lower-alpha 1]}. There are also various generalizations of measure to multifunctions.

Applications

Set-valued functions arise in optimal control theory, especially differential inclusions and related subjects as game theory, where the Kakutani fixed-point theorem for set-valued functions has been applied to prove existence of Nash equilibria. This among many other properties loosely associated with approximability of upper hemicontinuous multifunctions via continuous functions explains why upper hemicontinuity is more preferred than lower hemicontinuity.

Nevertheless, lower semi-continuous multifunctions usually possess continuous selections as stated in the Michael selection theorem, which provides another characterisation of paracompact spaces.^[6]^[7] Other selection theorems, like Bressan-Colombo directional continuous selection, Kuratowski and Ryll-Nardzewski measurable selection theorem, Aumann measurable selection, and Fryszkowski selection for decomposable maps are important in optimal control and the theory of differential inclusions.

Notes

↑ Some authors use the term ‘semicontinuous’ instead of ‘hemicontinuous’.

Related Research Articles

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function $mapping a nonempty compact convex set to itself, there is a point such that . The simplest forms of Brouwer's theorem are for continuous functions from a closed interval in the real numbers to itself or from a closed disk to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset of Euclidean space to itself.$

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.

<span class="mw-page-title-main">Functional analysis</span> Area of mathematics

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous or unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.

Topology is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

In mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for at least one point in its domain. It is a set-valued function with additional properties depending on context; some authors do not distinguish between set-valued functions and multifunctions, but English Wikipedia currently does, having a separate article for each.

Dušan D. Repovš is a Slovenian mathematician from Ljubljana, Slovenia.

<span class="mw-page-title-main">Closed graph theorem</span> Theorem relating continuity to graphs

In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous.

The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators.

In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations.

In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics.

In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of the Brouwer fixed point theorem. The Brouwer fixed point theorem is a fundamental result in topology which proves the existence of fixed points for continuous functions defined on compact, convex subsets of Euclidean spaces. Kakutani's theorem extends this to set-valued functions.

In mathematics, upper hemicontinuity and lower hemicontinuity are extensions of the notions of upper and lower semicontinuity of single-valued functions to set-valued functions. A set-valued function that is both upper and lower hemicontinuous is said to be continuous in an analogy to the property of the same name for single-valued functions.

In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form

In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following:

Ernest A. Michael was a prominent American mathematician known for his work in the field of general topology, most notably for his pioneering research on set-valued mappings. He is credited with developing the theory of continuous selections. The Michael selection theorem is named for him, which he proved in. Michael is also known in topology for the Michael line, a paracompact space whose product with the topological space of the irrational numbers is not normal. He wrote over 100 papers, mostly in the area of general topology.

The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters. The statement was first proven by Claude Berge in 1959. The theorem is primarily used in mathematical economics and optimal control.

In functional analysis, a branch of mathematics, a selection theorem is a theorem that guarantees the existence of a single-valued selection function from a given set-valued map. There are various selection theorems, and they are important in the theories of differential inclusions, optimal control, and mathematical economics.

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function $f : X \to Y$ between topological spaces has a closed graph if its graph is a closed subset of the product space $X \times Y$ . A related property is open graph.

In mathematics, particularly in functional analysis, the closed graph theorem is a result connecting the continuity of a linear operator to a topological property of their graph. Precisely, the theorem states that a linear operator between two Banach spaces is continuous if and only if the graph of the operator is closed.

References

↑ Aliprantis, Charalambos D.; Border, Kim C. (2013-03-14). Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer Science & Business Media. p. 523. ISBN 978-3-662-03961-8.
1 2 3 Wriggers, Peter; Panatiotopoulos, Panagiotis (2014-05-04). New Developments in Contact Problems. Springer. p. 29. ISBN 978-3-7091-2496-3.
↑ Repovš, Dušan (1998). Continuous selections of multivalued mappings. Pavel Vladimirovič. Semenov. Dordrecht: Kluwer Academic. ISBN 0-7923-5277-7. OCLC 39739641.
↑ Aumann, Robert J. (1965). "Integrals of Set-Valued Functions". Journal of Mathematical Analysis and Applications . 12 (1): 1–12. doi: 10.1016/0022-247X(65)90049-1 .
↑ Kakutani, Shizuo (1941). "A generalization of Brouwer's fixed point theorem". Duke Mathematical Journal . 8 (3): 457–459. doi:10.1215/S0012-7094-41-00838-4.
↑ Ernest Michael (Mar 1956). "Continuous Selections. I" (PDF). Annals of Mathematics. Second Series. 63 (2): 361–382. doi:10.2307/1969615. hdl:10338.dmlcz/119700. JSTOR 1969615.
↑ Dušan Repovš; P.V. Semenov (2008). "Ernest Michael and theory of continuous selections". Topology Appl. 155 (8): 755–763. arXiv: 0803.4473 . doi:10.1016/j.topol.2006.06.011. S2CID 14509315.

v t e Function
History List of specific functions
Types by domain and codomain	$X \to 𝔹$ $𝔹 \to X$ $𝔹ⁿ \to X$ $X \to ℤ$ $ℤ \to X$ $X \to ℝ$ $ℝ \to X$ $ℝⁿ \to X$ $X \to ℂ$ $ℂ \to X$ $ℂⁿ \to X$
Classes/properties	Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions	Restriction Composition λ Inverse
Generalizations	Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
Category