In mathematics, the arguments of the maxima (abbreviated arg max or argmax) are the points, or elements, of the domain of some function at which the function values are maximized.[note 1] In contrast to global maxima, which refers to the largest outputs of a function, arg max refers to the inputs, or arguments, at which the function outputs are as large as possible.
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. 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 .
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:
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.
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