Si_sinc.svg functions above have of {0} because both attain their global maximum value of 1 at x = 0. 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. ]] 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 X, a totally ordered set Y, and a function,, the over some subset S of X is defined by If S = X or S is clear from the context, then S is often left out, as in In other words, is the set of points x for which f(x) 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 f is identically equal to \infty on S 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 \sup {}_S f holds when f is not identically \infty on S.

Arg min

The notion of (or ), which stands for argument of the minimum, is defined analogously. For instance, are points x for which f(x) attains its smallest value. It is the complementary operator of. In the special case where are the extended real numbers, if f is identically equal to -\infty on S then (that is, ) and otherwise is defined as above and moreover, in this case (of f not identically equal to -\infty) it also satisfies:

Examples and properties

For example, if f(x) is 1 - |x|, then f attains its maximum value of 1 only at the point x = 0. Thus The operator is different from the \max operator. The \max 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 (in which case the maximum is undefined) or a singleton, but unlike may not contain multiple elements: for example, if f(x) is then but because the function attains the same value at every element of Equivalently, if M is the maximum of f, 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 { 5 }), since the maximum value of x (10 - x) is 25, which occurs for x = 5. However, in case the maximum is reached at many points, needs to be considered a of points. For example because the maximum value of \cos x is 1, which occurs on this interval for or 4 \pi. 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 x^3 is unbounded on the real line. As another example, although \arctan is bounded by \pm\pi/2. However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and thus a nonempty