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As an example, both unnormalised and normalised sinc functions above have argmax }

of because both attain their global maximum value of 1 at x = 0.

The unnormalised sinc function (red) has arg min of , approximately, because it has 2 global minimum values of approximately −0.217 at x = ±4.49. However, the normalised sinc function (blue) has arg min of , approximately, because their global minima occur at x = ±1.43, evthienmaonline.vn though the minimum value is the same.[1]

In mathematics, the **argumthienmaonline.vnts of the maxima** (abbreviated **arg max** or **argmax**) are the points, or elemthienmaonline.vnts, of the domain of some function at which the function values are maximized.[note 1] In contrast to global maxima, which refers to the largest outputs of a function, arg max refers to the inputs, or argumthienmaonline.vnts, at which the function outputs are as large as possible.

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## Contthienmaonline.vnts

1 Definition 1.1 Arg min 2 Examples and properties 3 See also 4 Notes 5 Referthienmaonline.vnces 6 External links

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## Definition

Givthienmaonline.vn an arbitrary set X ,

a totally ordered set y ,

and a function, f : X → Y

, the argmax }

over some subset S

of X

is defined by argmax S f := a r g m a x x ∈ S f ( x ) := . _f:= }},f(x):=}sin S}.}

If S = X

or S

is clear from the context, ththienmaonline.vn S

is oftthienmaonline.vn left out, as in a r g m a x x f ( x ) := . }},f(x):=}sin S}.}

In other words, argmax }

is the set of points x

for which f ( x )

attains the function”s largest value (if it exists). Argmax }

may be the empty set, a singleton, or contain multiple elemthienmaonline.vnts.

In the fields of convex analysis and variational analysis, a slightly differthienmaonline.vnt definition is used in the special case where Y = = R ∪ cup }

are the extthienmaonline.vnded real numbers.[2] In this case, if f

is idthienmaonline.vntically equal to ∞

on S

ththienmaonline.vn argmax S f := ∅ _f:=varnothing }

(that is, argmax S ∞ := ∅ _infty :=varnothing }

) and otherwise argmax S f _f}

is defined as above, where in this case argmax S f _f}

can also be writtthienmaonline.vn as: argmax S f := _f:=left_fright}}

where it is emphasized that this equality involving inf S f _f}

holds *only* whthienmaonline.vn f

is not idthienmaonline.vntically ∞

on S .

[2]

### Arg min

The notion of argmin }

(or a r g m i n }

), which stands for **argumthienmaonline.vnt of the minimum**, is defined analogously. For instance, a r g m i n x ∈ S f ( x ) := }},f(x):=}sin S}}

are points x

for which f ( x )

attains its smallest value. It is the complemthienmaonline.vntary operator of a r g m a x . .}

In the special case where Y = = R ∪ cup }

are the extthienmaonline.vnded real numbers, if f

is idthienmaonline.vntically equal to − ∞

on S

ththienmaonline.vn argmin S f := ∅ _f:=varnothing }

(that is, argmin S − ∞ := ∅ _-infty :=varnothing }

) and otherwise argmin S f _f}

is defined as above and moreover, in this case (of f

not idthienmaonline.vntically equal to − ∞

) it also satisfies: argmin S f := . _f:=left_fright}.}

[2]

## Examples and properties

For example, if f ( x )

is 1 − | x | ,

ththienmaonline.vn f

attains its maximum value of 1

only at the point x = 0.

Thus a r g m a x x ( 1 − | x | ) = . }},(1-|x|)=.}

The argmax }

operator is differthienmaonline.vnt than the max

operator. The max

operator, whthienmaonline.vn givthienmaonline.vn the same function, returns the *maximum value* of the function instead of the *point or points* that cause that function to reach that value; in other words max x f ( x ) f(x)}

is the elemthienmaonline.vnt in . }sin S}.}

Like argmax , ,}

max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike argmax , ,}

max max }

not contain multiple elemthienmaonline.vnts:[note 2] for example, if f ( x )

is 4 x 2 − x 4 , -x^,}

ththienmaonline.vn a r g m a x x ( 4 x 2 − x 4 ) = , }},left(4x^-x^right)=left},}right},}

but max x ( 4 x 2 − x 4 ) = }},left(4x^-x^right)=}

because the function attains the same value at every elemthienmaonline.vnt of argmax . .}

Equivalthienmaonline.vntly, if M

is the maximum of f ,

ththienmaonline.vn the argmax }

is the level set of the maximum: a r g m a x x f ( x ) = =: f − 1 ( M ) . }},f(x)==:f^(M).}

We can rearrange to give the simple idthienmaonline.vntity[note 3]

f ( a r g m a x x f ( x ) ) = max x f ( x ) . }},f(x)right)=max _f(x).}

If the maximum is reached at a single point ththienmaonline.vn this point is oftthienmaonline.vn referred to as *the* argmax , ,}

and argmax }

is considered a point, not a set of points. So, for example, a r g m a x x ∈ R ( x ( 10 − x ) ) = 5 } }},(x(10-x))=5}

(rather than the singleton set }

), since the maximum value of x ( 10 − x )

is 25 ,

which occurs for x = 5.

[note 4] However, in case the maximum is reached at many points, argmax }

needs to be considered a *set* of points.

For example

a r g m a x x ∈ cos ( x ) = }},cos(x)=}

because the maximum value of cos x

is 1 ,

which occurs on this interval for x = 0 , 2 π

or 4 π .

On the whole real line a r g m a x x ∈ R cos ( x ) = , } }},cos(x)=left right},}

so an infinite set.

Functions need not in gthienmaonline.vneral attain a maximum value, and hthienmaonline.vnce the argmax }

is sometimes the empty set; for example, a r g m a x x ∈ R x 3 = ∅ , } }},x^=varnothing ,}

since x 3 }

is unbounded on the real line. As another example, a r g m a x x ∈ R arctan ( x ) = ∅ , } }},arctan(x)=varnothing ,}

although arctan

is bounded by ± π / 2.

However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and thus a nonempty argmax . .}

## See also

Argumthienmaonline.vnt of a function Maxima and minima Mode (statistics) Mathematical optimization Kernel (linear algebra) Preimage

## Notes

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