Editing Arg max

{{Short description|Inputs at which function values are highest}}
{{refimprove|date=October 2014}}
[[File:Si_sinc.svg|thumb|350px|As an example, both unnormalised and normalised [[sinc]] functions above have <math>\operatorname{argmax}</math> of {0} because both attain their global maximum value of 1 at ''x''&nbsp;=&nbsp;0.<br /><br />The unnormalised sinc function (red) has ''arg min'' of {&minus;4.49,&nbsp;4.49}, approximately, because it has 2 global minimum values of approximately &minus;0.217 at ''x''&nbsp;=&nbsp;±4.49. However, the normalised sinc function (blue) has ''arg min'' of {&minus;1.43,&nbsp;1.43}, approximately, because their global minima occur at ''x''&nbsp;=&nbsp;±1.43, even though the minimum value is the same.<ref>"[http://physics.usyd.edu.au/teach_res/mp/doc/math_sinc_function.pdf The Unnormalized Sinc Function] {{Webarchive|url=https://web.archive.org/web/20170215045226/http://www.physics.usyd.edu.au/teach_res/mp/doc/math_sinc_function.pdf |date=2017-02-15 }}", University of Sydney</ref>]]

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 (mathematics)|function]] output value is [[Maxima and minima|maximized and minimized]], respectively.<ref group="note">For clarity, we refer to the input (''x'') as ''points'' and the output (''y'') as ''values;'' compare [[critical point (mathematics)|critical point]] and [[critical value (critical point)|critical value]].</ref> While the [[argument of a function|arguments]] are defined over the [[domain of a function]], the output is part of its [[codomain]].

== Definition ==

Given an arbitrary [[set (mathematics)|set]] {{nowrap|<math>X</math>,}} a [[totally ordered set]] {{nowrap|<math>Y</math>,}} and a function, {{nowrap|<math>f\colon X \to Y</math>,}} the <math>\operatorname{argmax}</math> over some subset <math>S</math> of <math>X</math> is defined by

:<math>\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 \}.</math>

If <math>S = X</math> or <math>S</math> is clear from the context, then <math>S</math> is often left out, as in <math>\underset{x}{\operatorname{arg\,max}}\, f(x) := \{ x ~:~ f(s) \leq f(x) \text{ for all } s \in X \}.</math> In other words, <math>\operatorname{argmax}</math> is the [[Set (mathematics)|set]] of points <math>x</math> for which <math>f(x)</math> attains the function's largest value (if it exists). <math>\operatorname{Argmax}</math> may be the [[empty set]], a [[Singleton (mathematics)|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 <math>Y = [-\infty,\infty] = \mathbb{R} \cup \{ \pm\infty \}</math> are the [[extended real numbers]].{{sfn|Rockafellar|Wets|2009|pp=1-37|ignore-err=yes}} In this case, if <math>f</math> is identically equal to <math>\infty</math> on <math>S</math> then <math>\operatorname{argmax}_S f := \varnothing</math> (that is, <math>\operatorname{argmax}_S \infty := \varnothing</math>) and otherwise <math>\operatorname{argmax}_S f</math> is defined as above, where in this case <math>\operatorname{argmax}_S f</math> can also be written as:
:<math>\operatorname{argmax}_S f := \left\{ x \in S ~:~ f(x) = \sup {}_S f \right\}</math> 

where it is emphasized that this equality involving <math>\sup {}_S f</math> holds {{em|only}} when <math>f</math> is not identically <math>\infty</math> on {{nowrap|<math>S</math>.}}{{sfn|Rockafellar|Wets|2009|pp=1-37|ignore-err=yes}} 

=== Arg min<!--'Arg min' redirects here--> ===
The notion of <math>\operatorname{argmin}</math> (or <math>\operatorname{arg\,min}</math>), which stands for '''argument of the minimum''', is defined analogously. For instance,

:<math>\underset{x \in S}{\operatorname{arg\,min}} \, f(x) := \{ x \in S ~:~ f(s) \geq f(x)  \text{ for all } s \in S \}</math>

are points <math>x</math> for which <math>f(x)</math> attains its smallest value. It is the complementary operator of {{nowrap|<math>\operatorname{arg\,max}</math>.}}

In the special case where <math>Y = [-\infty,\infty] = \R \cup \{ \pm\infty \}</math> are the [[extended real numbers]], if <math>f</math> is identically equal to <math>-\infty</math> on <math>S</math> then <math>\operatorname{argmin}_S f := \varnothing</math> (that is, <math>\operatorname{argmin}_S -\infty := \varnothing</math>) and otherwise <math>\operatorname{argmin}_S f</math> is defined as above and moreover, in this case (of <math>f</math> not identically equal to <math>-\infty</math>) it also satisfies:
:<math>\operatorname{argmin}_S f := \left\{ x \in S ~:~ f(x) = \inf {}_S f \right\}.</math>{{sfn|Rockafellar|Wets|2009|pp=1-37|ignore-err=yes}}

== Examples and properties ==

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

:<math>\underset{x}{\operatorname{arg\,max}}\, (1 - |x|) = \{ 0 \}.</math>

The <math>\operatorname{argmax}</math> operator is different from the <math>\max</math> operator. The <math>\max</math> operator, when given the same function, returns the {{em|[[Maxima and minima|maximum value]]}} of the function instead of the {{em|point or points}} that cause that function to reach that value; in other words

:<math>\max_x f(x)</math> is the element in <math>\{ f(x) ~:~ f(s) \leq f(x) \text{ for all } s \in S \}.</math>

Like <math>\operatorname{argmax},</math> max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike <math>\operatorname{argmax},</math> <math>\operatorname{max}</math> may not contain multiple elements:<ref group=note>Due to the [[Antisymmetric relation|anti-symmetry]] of <math>\,\leq,</math> a function can have at most one maximal value.</ref> for example, if <math>f(x)</math> is <math>4 x^2 - x^4,</math> then <math>\underset{x}{\operatorname{arg\,max}}\, \left( 4 x^2 - x^4 \right) = \left\{-\sqrt{2}, \sqrt{2}\right\},</math> but <math>\underset{x}{\operatorname{max}}\, \left( 4 x^2 - x^4 \right) = \{ 4 \}</math> because the function attains the same value at every element of <math>\operatorname{argmax}.</math>

Equivalently, if <math>M</math> is the maximum of <math>f,</math> then the <math>\operatorname{argmax}</math> is the [[level set]] of the maximum:

:<math>\underset{x}{\operatorname{arg\,max}} \, f(x) = \{ x ~:~ f(x) = M \} =: f^{-1}(M).</math>

We can rearrange to give the simple identity<ref group=note>This is an identity between sets, more particularly, between subsets of <math>Y.</math></ref>

:<math>f\left(\underset{x}{\operatorname{arg\,max}} \, f(x) \right) = \max_x f(x).</math>

If the maximum is reached at a single point then this point is often referred to as {{em|the}} <math>\operatorname{argmax},</math> and <math>\operatorname{argmax}</math> is considered a point, not a set of points. So, for example,

:<math>\underset{x\in\mathbb{R}}{\operatorname{arg\,max}}\, (x(10 - x)) = 5</math>

(rather than the [[Singleton (mathematics)|singleton]] set <math>\{ 5 \}</math>), since the maximum value of <math>x (10 - x)</math> is <math>25,</math> which occurs for <math>x = 5.</math><ref group="note">Note that <math>x (10 - x) = 25 - (x-5)^2 \leq 25</math> with equality if and only if <math>x - 5 = 0.</math></ref> However, in case the maximum is reached at many points, <math>\operatorname{argmax}</math> needs to be considered a {{em|set}} of points.

For example

:<math>\underset{x \in [0, 4 \pi]}{\operatorname{arg\,max}}\, \cos(x) = \{ 0, 2 \pi, 4 \pi \}</math>

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

:<math>\underset{x \in \mathbb{R}}{\operatorname{arg\,max}}\, \cos(x) = \left\{ 2 k \pi ~:~ k \in \mathbb{Z} \right\},</math> so an infinite set.

Functions need not in general attain a maximum value, and hence the <math>\operatorname{argmax}</math> is sometimes the [[empty set]]; for example, <math>\underset{x\in\mathbb{R}}{\operatorname{arg\,max}}\, x^3 = \varnothing,</math> since <math>x^3</math> is [[Bounded function|unbounded]] on the real line. As another example, <math>\underset{x \in \mathbb{R}}{\operatorname{arg\,max}}\, \arctan(x) = \varnothing,</math> although [[Arc tangent|<math>\arctan</math>]] is bounded by <math>\pm\pi/2.</math> However, by the [[extreme value theorem]], a continuous real-valued function on a [[Interval (mathematics)|closed interval]] has a maximum, and thus a nonempty <math>\operatorname{argmax}.</math>

==See also==

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

==Notes==
{{reflist|group=note}}

==References==
{{reflist}}
* {{Rockafellar Wets Variational Analysis 2009 Springer}} <!-- {{sfn|Rockafellar|Wets|2009|p=}} -->

==External links==
*{{PlanetMath|urlname=argminandargmax|title=arg min and arg max}}

[[Category:Elementary mathematics]]
[[Category:Inverse functions]]