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. 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.
Definition
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. 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 when is not identically on
Arg min
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:
Examples and properties
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 of the function instead of the 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: 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
If the maximum is reached at a single point then this point is often referred to as 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 However, in case the maximum is reached at many points, needs to be considered a 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
Argument of a function
Maxima and minima
Mode (statistics) Mathematical optimization Kernel (linear algebra) Preimage