Arg max

Examples and properties

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

${\displaystyle \underset{x}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}(1-|x|)=\{0\}.}$

The ${\displaystyle \mathrm{argmax}}$ operator is different from the ${\displaystyle max}$ operator. The ${\displaystyle max}$ 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

${\displaystyle \underset{x}{max}f(x)}$ is the element in ${\displaystyle \{f(x):f(s)\le f(x)\text{ for all }s\in S\}.}$

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

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

${\displaystyle \underset{x}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}f(x)=\{x:f(x)=M\}=:f^{-1}(M).}$

We can rearrange to give the simple identity

${\displaystyle f\left(\underset{x}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}f(x)\right)=\underset{x}{max}f(x).}$

If the maximum is reached at a single point then this point is often referred to as the ${\displaystyle \mathrm{argmax},}$ and ${\displaystyle \mathrm{argmax}}$ is considered a point, not a set of points. So, for example,

${\displaystyle \underset{x\in \mathbb{R}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}(x(10-x))=5}$

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

For example

${\displaystyle \underset{x\in [0,4\pi ]}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}\cos (x)=\{0,2\pi ,4\pi \}}$

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

${\displaystyle \underset{x\in \mathbb{R}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}\cos (x)=\left\{2k\pi :k\in \mathbb{Z}\right\},}$ so an infinite set.

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