A set-valued function (or correspondence) is a mathematical function that maps elements from one set, the domain of the function, to subsets of another set. Set-valued functions are used in a variety of mathematical fields, including

optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...

and game theory. Set-valued functions are also known as multivalued functions in some references, but herein and in many others references in

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

, a

multivalued function In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...

is a set-valued function 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 close to , and thus defines

locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). P ...

an ordinary function. Multivalued_function

Examples

The argmax of a function is in general, multivalued. For example,

\operatorname_  \cos(x) = \

Set-valued analysis

Set-valued analysis is the study of sets in the spirit of

and

general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometri ...

. 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 Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of som ...

; the term "

variational analysis In mathematics, the term variational analysis usually denotes the combination and extension of methods from convex optimization and the classical calculus of variations to a more general theory. This includes the more general problems of optimizati ...

" is used by authors such as R. Tyrrell Rockafellar and Roger J-B Wets,

Jonathan Borwein Jonathan Michael Borwein (20 May 1951 – 2 August 2016) was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. He was a close associate of David H. Bailey, and they ...

and

Adrian Lewis Adrian Lewis (born 21 January 1985) is an English professional darts player currently playing in the PDC. He is a two-time PDC World Darts Champion, winning in 2011 and 2012. He is nicknamed Jackpot, as he won a jackpot gambling in Las Vegas ...

, 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, implicit function theorem, contraction mappings, measure theory,

fixed-point theorems In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors cla ...

, and

topological degree theory In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solutio ...

. 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. There are also various generalizations of measure to multifunctions.

Applications

Set-valued functions arise in

optimal control theory Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

, especially differential inclusions and related subjects as game theory, where the

Kakutani fixed-point theorem 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 set, convex, compact set, compact subset of a Euclidean sp ...

for set-valued functions has been applied to prove existence of

Nash equilibria In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equili ...

. 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 In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...

spaces. Other selection theorems, like Bressan-Colombo directional continuous selection,

Kuratowski and Ryll-Nardzewski measurable selection theorem In mathematics, the Kuratowski–Ryll-Nardzewski measurable selection theorem is a result from measure theory that gives a sufficient condition for a set-valued function to have a measurable selection function. It is named after the Polish mathem ...

, Aumann measurable selection, and Fryszkowski selection for decomposable maps are important in optimal control and the theory of differential inclusions.

Examples

Set-valued analysis

Applications

Notes

References

Further reading

See also