As an example, both unnormalised and normalised sinc functions above have argmax {\displaystyle \operatorname {argmax} } of {0} because both attain their global maximum value of 1 at x = 0. The unnormalised sinc function (red) has arg min of {−4.49, 4.49}, 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 {−1.43, 1.43}, approximately, because their global minima occur at x = ±1.43, even though the minimum value is the same.

In mathematics, the arguments of the maxima (abbreviated arg max or argmax) and arguments of the minima (abbreviated arg min or argmin) are the input points at which a function output value is maximized and minimized, respectively. While the arguments are defined over the domain of a function, the output is part of its codomain.

Definition

Given an arbitrary set X {\displaystyle X}, a totally ordered set Y {\displaystyle Y}, and a function, f : X → Y {\displaystyle f\colon X\to Y}, the argmax {\displaystyle \operatorname {argmax} } over some subset S {\displaystyle S} of X {\displaystyle X} is defined by

argmax S ⁡ f := a r g m a x x ∈ S f ( x ) := { x ∈ S : f ( s ) ≤ f ( x ) for all s ∈ S } . {\displaystyle \operatorname {argmax} _{S}f:={\underset {x\in S}{\operatorname {arg\,max} }}\,f(x):=\{x\in S~:~f(s)\leq f(x){\text{ for all }}s\in S\}.}

If S = X {\displaystyle S=X} or S {\displaystyle S} is clear from the context, then S {\displaystyle S} is often left out, as in a r g m a x x f ( x ) := { x : f ( s ) ≤ f ( x ) for all s ∈ X } . {\displaystyle {\underset {x}{\operatorname {arg\,max} }}\,f(x):=\{x~:~f(s)\leq f(x){\text{ for all }}s\in X\}.} In other words, argmax {\displaystyle \operatorname {argmax} } is the set of points x {\displaystyle x} for which f ( x ) {\displaystyle f(x)} attains the function's largest value (if it exists). Argmax {\displaystyle \operatorname {Argmax} } may be the empty set, a singleton, or contain multiple elements.

In the fields of convex analysis and variational analysis, a slightly different definition is used in the special case where Y = [ − ∞ , ∞ ] = R ∪ { ± ∞ } {\displaystyle Y=[-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}} are the extended real numbers. In this case, if f {\displaystyle f} is identically equal to ∞ {\displaystyle \infty } on S {\displaystyle S} then argmax S ⁡ f := ∅ {\displaystyle \operatorname {argmax} _{S}f:=\varnothing } (that is, argmax S ⁡ ∞ := ∅ {\displaystyle \operatorname {argmax} _{S}\infty :=\varnothing }) and otherwise argmax S ⁡ f {\displaystyle \operatorname {argmax} _{S}f} is defined as above, where in this case argmax S ⁡ f {\displaystyle \operatorname {argmax} _{S}f} can also be written as:

argmax S ⁡ f := { x ∈ S : f ( x ) = sup S f } {\displaystyle \operatorname {argmax} _{S}f:=\left\{x\in S~:~f(x)=\sup {}_{S}f\right\}}

where it is emphasized that this equality involving sup S f {\displaystyle \sup {}_{S}f} holds only when f {\displaystyle f} is not identically ∞ {\displaystyle \infty } on S {\displaystyle S}.

Arg min

The notion of argmin {\displaystyle \operatorname {argmin} } (or a r g m i n {\displaystyle \operatorname {arg\,min} }), which stands for argument of the minimum, is defined analogously. For instance,

a r g m i n x ∈ S f ( x ) := { x ∈ S : f ( s ) ≥ f ( x ) for all s ∈ S } {\displaystyle {\underset {x\in S}{\operatorname {arg\,min} }}\,f(x):=\{x\in S~:~f(s)\geq f(x){\text{ for all }}s\in S\}}

are points x {\displaystyle x} for which f ( x ) {\displaystyle f(x)} attains its smallest value. It is the complementary operator of a r g m a x {\displaystyle \operatorname {arg\,max} }.

In the special case where Y = [ − ∞ , ∞ ] = R ∪ { ± ∞ } {\displaystyle Y=[-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}} are the extended real numbers, if f {\displaystyle f} is identically equal to − ∞ {\displaystyle -\infty } on S {\displaystyle S} then argmin S ⁡ f := ∅ {\displaystyle \operatorname {argmin} _{S}f:=\varnothing } (that is, argmin S − ∞ := ∅ {\displaystyle \operatorname {argmin} _{S}-\infty :=\varnothing }) and otherwise argmin S ⁡ f {\displaystyle \operatorname {argmin} _{S}f} is defined as above and moreover, in this case (of f {\displaystyle f} not identically equal to − ∞ {\displaystyle -\infty }) it also satisfies:

argmin S ⁡ f := { x ∈ S : f ( x ) = inf S f } . {\displaystyle \operatorname {argmin} _{S}f:=\left\{x\in S~:~f(x)=\inf {}_{S}f\right\}.}

Examples and properties

For example, if f ( x ) {\displaystyle f(x)} is 1 − | x | , {\displaystyle 1-|x|,} then f {\displaystyle f} attains its maximum value of 1 {\displaystyle 1} only at the point x = 0. {\displaystyle x=0.} Thus

a r g m a x x ( 1 − | x | ) = { 0 } . {\displaystyle {\underset {x}{\operatorname {arg\,max} }}\,(1-|x|)=\{0\}.}

The argmax {\displaystyle \operatorname {argmax} } operator is different from the max {\displaystyle \max } operator. The max {\displaystyle \max } operator, when given 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 ) {\displaystyle \max _{x}f(x)} is the element in { f ( x ) : f ( s ) ≤ f ( x ) for all s ∈ S } . {\displaystyle \{f(x)~:~f(s)\leq f(x){\text{ for all }}s\in S\}.}

Like argmax , {\displaystyle \operatorname {argmax} ,} max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike argmax , {\displaystyle \operatorname {argmax} ,} max {\displaystyle \operatorname {max} } may not contain multiple elements: for example, if f ( x ) {\displaystyle f(x)} is 4 x 2 − x 4 , {\displaystyle 4x^{2}-x^{4},} then a r g m a x x ( 4 x 2 − x 4 ) = { − 2 , 2 } , {\displaystyle {\underset {x}{\operatorname {arg\,max} }}\,\left(4x^{2}-x^{4}\right)=\left\{-{\sqrt {2}},{\sqrt {2}}\right\},} but max x ( 4 x 2 − x 4 ) = { 4 } {\displaystyle {\underset {x}{\operatorname {max} }}\,\left(4x^{2}-x^{4}\right)=\{4\}} because the function attains the same value at every element of argmax . {\displaystyle \operatorname {argmax} .}

Equivalently, if M {\displaystyle M} is the maximum of f , {\displaystyle f,} then the argmax {\displaystyle \operatorname {argmax} } is the level set of the maximum:

a r g m a x x f ( x ) = { x : f ( x ) = M } =: f − 1 ( M ) . {\displaystyle {\underset {x}{\operatorname {arg\,max} }}\,f(x)=\{x~:~f(x)=M\}=:f^{-1}(M).}

We can rearrange to give the simple identity

f ( a r g m a x x f ( x ) ) = max x f ( x ) . {\displaystyle f\left({\underset {x}{\operatorname {arg\,max} }}\,f(x)\right)=\max _{x}f(x).}

If the maximum is reached at a single point then this point is often referred to as the argmax , {\displaystyle \operatorname {argmax} ,} and argmax {\displaystyle \operatorname {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 {\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {arg\,max} }}\,(x(10-x))=5}

(rather than the singleton set { 5 } {\displaystyle \{5\}}), since the maximum value of x ( 10 − x ) {\displaystyle x(10-x)} is 25 , {\displaystyle 25,} which occurs for x = 5. {\displaystyle x=5.} However, in case the maximum is reached at many points, argmax {\displaystyle \operatorname {argmax} } needs to be considered a set of points.

For example

a r g m a x x ∈ [ 0 , 4 π ] cos ⁡ ( x ) = { 0 , 2 π , 4 π } {\displaystyle {\underset {x\in [0,4\pi ]}{\operatorname {arg\,max} }}\,\cos(x)=\{0,2\pi ,4\pi \}}

because the maximum value of cos ⁡ x {\displaystyle \cos x} is 1 , {\displaystyle 1,} which occurs on this interval for x = 0 , 2 π {\displaystyle x=0,2\pi } or 4 π . {\displaystyle 4\pi .} On the whole real line

a r g m a x x ∈ R cos ⁡ ( x ) = { 2 k π : k ∈ Z } , {\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {arg\,max} }}\,\cos(x)=\left\{2k\pi ~:~k\in \mathbb {Z} \right\},} so an infinite set.

Functions need not in general attain a maximum value, and hence the argmax {\displaystyle \operatorname {argmax} } is sometimes the empty set; for example, a r g m a x x ∈ R x 3 = ∅ , {\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {arg\,max} }}\,x^{3}=\varnothing ,} since x 3 {\displaystyle x^{3}} is unbounded on the real line. As another example, a r g m a x x ∈ R arctan ⁡ ( x ) = ∅ , {\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {arg\,max} }}\,\arctan(x)=\varnothing ,} although arctan {\displaystyle \arctan } is bounded by ± π / 2. {\displaystyle \pm \pi /2.} However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and thus a nonempty argmax . {\displaystyle \operatorname {argmax} .}

See also

Notes

External links