Arg max
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.
Given an arbitrary set
, a totally ordered set
is the set of points
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.
is identically equal to
can also be written as: where it is emphasized that this equality involving
), which stands for argument of the minimum, is defined analogously.
For instance, are points
attains its smallest value.
It is the complementary operator of
In the special case where
are the extended real numbers, if
attains its maximum value of
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 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
because the function attains the same value at every element of
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
So, for example, (rather than the singleton set
[note 4] However, in case the maximum is reached at many points,
needs to be considered a set of points.
On the whole real line Functions need not in general attain a maximum value, and hence the
is sometimes the empty set; for example,
is unbounded on the real line.
However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and thus a nonempty
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 ]