Aset-valued function, also called acorrespondence orset-valuedrelation, is a mathematicalfunction that maps elements from one set, thedomain of the function, to subsets of another set.[1][2] Set-valued functions are used in a variety of mathematical fields, includingoptimization,control theory andgame theory.
Set-valued functions are also known asmultivalued functions in some references,[3] but this article and the articleMultivalued function follow the authors who make a distinction.
Although other authors may distinguish them differently (or not at all), Wriggers and Panatiotopoulos (2014) distinguish multivalued functions from set-valued functions (which they calledset-valued relations) by the fact that multivalued functions only take multiple values at finitely (or denumerably) many points, and otherwise behave like afunction.[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, amultivalued function is a set-valued functionf that has a furthercontinuity property, namely that the choice of an element in the set defines a corresponding element in each set fory close tox, and thus defineslocally an ordinary function.
Theargmax of a function is in general, multivalued. For example,.
Set-valued analysis is the study of sets in the spirit ofmathematical analysis andgeneral 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 ofmathematical economics andoptimal control, partly as a generalization ofconvex analysis; the term "variational analysis" is used by authors such asR. Tyrrell Rockafellar andRoger J-B Wets,Jonathan Borwein andAdrian Lewis, andBoris Mordukhovich. In optimization theory, the convergence of approximatingsubdifferentials 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, andtopological degree theory. In particular,equations are generalized toinclusions, while differential equations are generalized todifferential inclusions.
One can distinguish multiple concepts generalizingcontinuity, such as theclosed graph property andupper and lower hemicontinuity[a]. There are also various generalizations ofmeasure to multifunctions.
Set-valued functions arise inoptimal control theory, especiallydifferential inclusions and related subjects asgame theory, where theKakutani fixed-point theorem for set-valued functions has been applied to prove existence ofNash 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 theMichael selection theorem, which provides another characterisation ofparacompact 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 inoptimal control and the theory ofdifferential inclusions.
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