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

Definition

Given an arbitrary set a totally ordered set and a function, is defined by

is the set may be the empty set, a singleton, or contain multiple elements.

In the fields of convex analysis and variational analysis are the extended real numbers. can also be written as:

holds on

Arg min

), which stands for argument of the minimum, is defined analogously. For instance,

attains its smallest value. It is the complementary operator of

are the extended real numbers) it also satisfies:

Examples and properties

Thus

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

may not contain multiple elements:

is the level set of the maximum:

We can rearrange to give the simple identity

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

(rather than the singleton needs to be considered a of points.

For example

On the whole real line

1. so an infinite set.

is sometimes the empty set is unbounded on the real line although \arctan However, by the extreme value theorem, a continuous real-valued function on a closed interval

See also

• Argument of a function

• Maxima and minima

• Mode (statistics)

• Mathematical optimization

• Kernel (linear algebra)

• Preimage

• Softmax function

Notes

References

External links