Fungrim entry: 0a3e5a

Symbol: ArgMin —

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

— Locations of minimum value

ArgMin(f(x), x, P(x)), rendered

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

, if the minimum value exists.

f\!\left(x\right)

does not attain a minimum 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
`ArgMin`	$\mathop{\operatorname{arg\,min}}\limits_{P\left(x\right)} f\!\left(x\right)$	Locations of minimum value
`Minimum`	$\mathop{\min}\limits_{P\left(x\right)} f\!\left(x\right)$	Minimum value of a set or function

Source code for this entry:

Entry(ID("0a3e5a"),
    SymbolDefinition(ArgMin, ArgMin(f(x), x, P(x)), "Locations of minimum value"),
    Description(SourceForm(ArgMin(f(x), x, P(x))), ", rendered", ArgMin(f(x), x, P(x)), ", gives the set of points", r, "satisfying", P(r), "such that", Equal(f(r), Minimum(f(x), x, P(x))), ", if the minimum value exists."),
    Description("If", f(x), "does not attain a minimum 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