Argmax

In mathematics, the arguments of the maxima (abbreviated arg max or argmax) are the points, or elements, of the domain of some function at which the function values are maximized.For clarity, we refer to the input (''x'') as ''points'' and the output (''y'') as ''values;'' compare critical point and critical value. In contrast to global maxima, which refers to the largest ''outputs'' of a function, arg max refers to the ''inputs'', or arguments, at which the function outputs are as large as possible.

Definition

Given an arbitrary set a totally ordered set and a function, the $\operatorname$ over some subset $S$ of $X$ is defined by

:$\operatorname_S f := \underset\, f(x) := \.$

If $S = X$ or $S$ is clear from the context, then $S$ is often left out, as in $\underset\, f(x) := \.$ In other words, $\operatorname$ is the set of points $x$ for which $f(x)$ attains the function's largest value (if it exists). $\operatorname$ 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 = \infty,\infty= \mathbb \cup \ are the extended real numbers. In this case, if $f$ is identically equal to $\infty$ on $S$ then $\operatorname_S f := \varnothing$ (that is, $\operatorname_S \infty := \varnothing$) and otherwise $\operatorname_S f$ is defined as above, where in this case $\operatorname_S f$ can also be written as:
:$\operatorname_S f := \left\$

where it is emphasized that this equality involving $\sup _S f$ holds when $f$ is not identically $\infty$ on

Arg min

The notion of $\operatorname$ (or $\operatorname$), which stands for argument of the minimum, is defined analogously. For instance,

:$\underset \, f(x) := \$

are points $x$ for which $f(x)$ attains its smallest value. It is the complementary operator of

In the special case where Y = \infty,\infty= \R \cup \ are the extended real numbers, if $f$ is identically equal to $-\infty$ on $S$ then $\operatorname_S f := \varnothing$ (that is, $\operatorname_S -\infty := \varnothing$) and otherwise $\operatorname_S f$ is defined as above and moreover, in this case (of $f$ not identically equal to $-\infty$) it also satisfies:
:$\operatorname_S f := \left\.$

Examples and properties

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

:$\underset\, (1 - , x, ) = \.$

The $\operatorname$ operator is different from the $\max$ operator. The $\max$ operator, when given the same function, returns the of the function instead of the that cause that function to reach that value; in other words

:$\max_x f(x)$ is the element in $\.$

Like $\operatorname,$ max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike $\operatorname,$ $\operatorname$ may not contain multiple elements:Due to the anti-symmetry of $\,\leq,$ a function can have at most one maximal value. for example, if $f(x)$ is $4 x^2 - x^4,$ then $\underset\, \left( 4 x^2 - x^4 \right) = \left\,$ but $\underset\, \left( 4 x^2 - x^4 \right) = \$ because the function attains the same value at every element of $\operatorname.$

Equivalently, if $M$ is the maximum of $f,$ then the $\operatorname$ is the level set of the maximum:

:$\underset \, f(x) = \ =: f^(M).$

We can rearrange to give the simple identityThis is an identity between sets, more particularly, between subsets of $Y.$

:$f\left(\underset \, f(x) \right) = \max_x f(x).$

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

:$\underset\, (x(10 - x)) = 5$

(rather than the singleton set $\$), since the maximum value of $x (10 - x)$ is $25,$ which occurs for $x = 5.$Note that $x (10 - x) = 25 - (x-5)^2 \leq 25$ with equality if and only if $x - 5 = 0.$ However, in case the maximum is reached at many points, $\operatorname$ needs to be considered a of points.

For example

:$\underset\, \cos(x) = \$

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

:$\underset\, \cos(x) = \left\,$ so an infinite set.

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

See also

* Argument of a function
* Maxima and minima
* Mode (statistics)
* Mathematical optimization
* Kernel (linear algebra)
* Preimage

Notes

References

*

External links

*{{PlanetMath, urlname=argminandargmax, title=arg min and arg max

Elementary mathematics
Inverse functions