Fungrim entry: 617fe3

Symbol: ArgMax —

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

— Locations of maximum value

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

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

, gives the set of values

x \in S

such that

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

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

\mathop{\operatorname{arg\,max}}\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{\max}\limits_{s \in S,\,P(s)} f(s)

f(x)

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

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

\mathop{\operatorname{arg\,max}}\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{\max}\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
`ArgMax`	$\mathop{\operatorname{arg\,max}}\limits_{x \in S} f(x)$	Locations of maximum value
`Maximum`	$\mathop{\max}\limits_{x \in S} f(x)$	Maximum value of a set or function

Source code for this entry:

Entry(ID("617fe3"),
    SymbolDefinition(ArgMax, ArgMax(f(x), ForElement(x, S)), "Locations of maximum value"),
    Description(SourceForm(ArgMax(f(x), ForElement(x, S))), ", rendered", ArgMax(f(x), ForElement(x, S)), ", gives the set of values", Element(x, S), "such that", Equal(f(x), Maximum(f(s), ForElement(s, S))), "."),
    Description(SourceForm(ArgMax(f(x), ForElement(x, S), P(x))), ", rendered", ArgMax(f(x), ForElement(x, S), P(x)), ", gives the set of values", Element(x, S), "satisfying", P(x), "and such that", Equal(f(x), Maximum(f(s), ForElement(s, S), P(s))), "."),
    Description("If", f(x), "does not attain a maximum value satisfying the conditions, the result is the empty set", Set(), "."),
    Description(SourceForm(ArgMax(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", ArgMax(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), Maximum(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