Arg max
In mathematics, the arguments of the maxima and arguments of the minima 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, the over some subset of is defined byIf 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. 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 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, 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 and otherwise is defined as above and moreover, in this case it also satisfies:
Examples and properties
For example, if is then attains its maximum value of only at the point ThusThe 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
Like max may be the empty set 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,
, 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
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 [Arc tangent|] 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