Arg max

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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.[note 1] While the arguments are defined over the domain of a function, the output is part of its codomain.

As an example, both unnormalised and normalised sinc functions above have 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.[1]

Definition

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Given an arbitrary set  , a totally ordered set  , and a function,  , the   over some subset   of   is defined by

 

If   or   is clear from the context, then   is often left out, as in   In other words,   is the set of points   for which   attains the function's largest value (if it exists).   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   are the extended real numbers.[2] In this case, if   is identically equal to   on   then   (that is,  ) and otherwise   is defined as above, where in this case   can also be written as:

 

where it is emphasized that this equality involving   holds only when   is not identically   on  .[2]

Arg min

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

 

are points   for which   attains its smallest value. It is the complementary operator of  .

In the special case where   are the extended real numbers, if   is identically equal to   on   then   (that is,  ) and otherwise   is defined as above and moreover, in this case (of   not identically equal to  ) it also satisfies:

 [2]

Examples and properties

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For example, if   is   then   attains its maximum value of   only at the point   Thus

 

The   operator is different from the   operator. The   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

  is the element in  

Like   max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike     may not contain multiple elements:[note 2] for example, if   is   then   but   because the function attains the same value at every element of  

Equivalently, if   is the maximum of   then the   is the level set of the maximum:

 

We can rearrange to give the simple identity[note 3]

 

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

 

(rather than the singleton set  ), since the maximum value of   is   which occurs for  [note 4] However, in case the maximum is reached at many points,   needs to be considered a set of points.

For example

 

because the maximum value of   is   which occurs on this interval for   or   On the whole real line

  so an infinite set.

Functions need not in general attain a maximum value, and hence the   is sometimes the empty set; for example,   since   is unbounded on the real line. As another example,   although   is bounded by   However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and thus a nonempty  

See also

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Notes

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  1. ^ For clarity, we refer to the input (x) as points and the output (y) as values; compare critical point and critical value.
  2. ^ Due to the anti-symmetry of   a function can have at most one maximal value.
  3. ^ This is an identity between sets, more particularly, between subsets of  
  4. ^ Note that   with equality if and only if  

References

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  1. ^ "The Unnormalized Sinc Function Archived 2017-02-15 at the Wayback Machine", University of Sydney
  2. ^ a b c Rockafellar & Wets 2009, pp. 1–37.
  • Rockafellar, R. Tyrrell; Wets, Roger J.-B. (26 June 2009). Variational Analysis. Grundlehren der mathematischen Wissenschaften. Vol. 317. Berlin New York: Springer Science & Business Media. ISBN 9783642024313. OCLC 883392544.
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