Fungrim entry: 617fe3

Symbol: ArgMax —

\mathop{\operatorname{arg\,max}}\limits_{P\left(x\right)} f\!\left(x\right)

— Locations of maximum value

ArgMax(f(x), x, P(x)), rendered

\mathop{\operatorname{arg\,max}}\limits_{P\left(x\right)} f\!\left(x\right)

, gives the set of points

r

satisfying

P\!\left(r\right)

such that

f\!\left(r\right) = \mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right)

, if the maximum value exists.

f\!\left(x\right)

does not attain a maximum value on the set of points defined by

P\!\left(x\right)

, the result is the empty set

\left\{\right\}

The argument x to this operator defines a locally bound variable. The corresponding predicate

P\!\left(x\right)

must define the domain of

x

unambiguously; that is, it must include a statement such as

x \in S

where

S

is a known set. More generally, x can be a collection of variables

\left(x, y, \ldots\right)

all of which become locally bound, with a corresponding predicate

P\!\left(x, y, \ldots\right)

Definitions:

Fungrim symbol	Notation	Short description
`ArgMax`	$\mathop{\operatorname{arg\,max}}\limits_{P\left(x\right)} f\!\left(x\right)$	Locations of maximum value
`Maximum`	$\mathop{\max}\limits_{P\left(x\right)} f\!\left(x\right)$	Maximum value of a set or function

Source code for this entry:

Entry(ID("617fe3"),
    SymbolDefinition(ArgMax, ArgMax(f(x), x, P(x)), "Locations of maximum value"),
    Description(SourceForm(ArgMax(f(x), x, P(x))), ", rendered", ArgMax(f(x), x, P(x)), ", gives the set of points", r, "satisfying", P(r), "such that", Equal(f(r), Maximum(f(x), x, P(x))), ", if the maximum value exists."),
    Description("If", f(x), "does not attain a maximum value on the set of points defined by", P(x), ", the result is the empty set", Set(), "."),
    Description("The argument", SourceForm(x), "to this operator defines a locally bound variable.", "The corresponding predicate", P(x), "must define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set.", "More generally,", SourceForm(x), "can be a collection of variables", Tuple(x, y, Ellipsis), "all of which become locally bound, with a corresponding predicate", P(x, y, Ellipsis), "."))

Topics using this entry

Operators