Fungrim entry: 0a3e5a

Symbol: ArgMin —

\mathop{\operatorname{arg\,min}}\limits_{x \in S} f(x)

— Locations of minimum value

ArgMin(f(x), ForElement(x, S)), rendered

\mathop{\operatorname{arg\,min}}\limits_{x \in S} f(x)

, gives the set of values

x \in S

such that

f(x) = \mathop{\min}\limits_{s \in S} f(s)

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

\mathop{\operatorname{arg\,min}}\limits_{x \in S,\,P(x)} f(x)

, gives the set of values

x \in S

satisfying

P(x)

and such that

f(x) = \mathop{\min}\limits_{s \in S,\,P(s)} f(s)

f(x)

does not attain a minimum value satisfying the conditions, the result is the empty set {}.

ArgMin(f(x, y), For(Tuple(x, y)), P(x, y)), rendered

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

, gives the set of tuples

\left(x, y\right)

satisfying

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

such that

f\!\left(x, y\right) = \mathop{\min}\limits_{P\left(s, t\right)} f\!\left(s, t\right)

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

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. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

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

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

, etc.

Definitions:

Fungrim symbol	Notation	Short description
`ArgMin`	$\mathop{\operatorname{arg\,min}}\limits_{x \in S} f(x)$	Locations of minimum value
`Minimum`	$\mathop{\min}\limits_{x \in S} f(x)$	Minimum value of a set or function

Source code for this entry:

Entry(ID("0a3e5a"),
    SymbolDefinition(ArgMin, ArgMin(f(x), ForElement(x, S)), "Locations of minimum value"),
    Description(SourceForm(ArgMin(f(x), ForElement(x, S))), ", rendered", ArgMin(f(x), ForElement(x, S)), ", gives the set of values", Element(x, S), "such that", Equal(f(x), Minimum(f(s), ForElement(s, S))), "."),
    Description(SourceForm(ArgMin(f(x), ForElement(x, S), P(x))), ", rendered", ArgMin(f(x), ForElement(x, S), P(x)), ", gives the set of values", Element(x, S), "satisfying", P(x), "and such that", Equal(f(x), Minimum(f(s), ForElement(s, S), P(s))), "."),
    Description("If", f(x), "does not attain a minimum value satisfying the conditions, the result is the empty set", Set(), "."),
    Description(SourceForm(ArgMin(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", ArgMin(f(x, y), For(Tuple(x, y)), P(x, y)), ", gives the set of tuples", Tuple(x, y), "satisfying", P(x, y), "such that", Equal(f(x, y), Minimum(f(s, t), For(Tuple(s, t)), P(s, t))), ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", 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. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

Topics using this entry

Operators