Editing Arg max (section)

== 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}}