arg max#Arg min

{{Short description|Inputs at which function values are highest}}

{{refimprove|date=October 2014}}

File:Si_sinc.svg functions above have \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."[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]]

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.For clarity, we refer to the input (x) as points and the output (y) as values; compare critical point and critical value. While the arguments are defined over the domain of a function, the output is part of its codomain.

Definition

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

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

:\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 holds {{em|only}} when f is not identically \infty on {{nowrap|S.}}{{sfn|Rockafellar|Wets|2009|pp=1-37|ignore-err=yes}}

= Arg min<!--'Arg min' redirects here--> =

The notion of \operatorname{argmin} (or \operatorname{arg\,min}), which stands for argument of the minimum, is defined analogously. For instance,

:\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 for which f(x) attains its smallest value. It is the complementary operator of {{nowrap|\operatorname{arg\,max}.}}

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

:\operatorname{argmin}_S f := \left\{ x \in S ~:~ f(x) = \inf {}_S f \right\}.{{sfn|Rockafellar|Wets|2009|pp=1-37|ignore-err=yes}}

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{x}{\operatorname{arg\,max}}\, (1 - |x|) = \{ 0 \}.

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

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

Like \operatorname{argmax}, max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike \operatorname{argmax}, \operatorname{max} 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{x}{\operatorname{arg\,max}}\, \left( 4 x^2 - x^4 \right) = \left\{-\sqrt{2}, \sqrt{2}\right\}, but \underset{x}{\operatorname{max}}\, \left( 4 x^2 - x^4 \right) = \{ 4 \} because the function attains the same value at every element of \operatorname{argmax}.

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

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

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

: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 {{em|the}} \operatorname{argmax}, and \operatorname{argmax} is considered a point, not a set of points. So, for example,

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

(rather than the singleton set \{ 5 \}), 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{argmax} needs to be considered a {{em|set}} of points.

For example

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

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

Functions need not in general attain a maximum value, and hence the \operatorname{argmax} is sometimes the empty set; for example, \underset{x\in\mathbb{R}}{\operatorname{arg\,max}}\, x^3 = \varnothing, since x^3 is unbounded on the real line. As another example, \underset{x \in \mathbb{R}}{\operatorname{arg\,max}}\, \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{argmax}.

See also

Notes

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References

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  • {{Rockafellar Wets Variational Analysis 2009 Springer}}