In mathematics, the arguments of the maxima 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 arbitrarysetX, a totally ordered setY, and a function,, the arg max 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, arg max is the set of points, x, for which f attains the function's largest value. Arg max may be the empty set, a singleton, or contain multiple elements. For example, if f is 1−|x|, then f attains its maximum value of 1 only at the point x = 0. Thus, The arg max operator is different than the max operator. The max operator, when given the same function, returns the maximum value instead of the point or points that reach that value; in other words Like arg max, max may be the empty set or a singleton, but unlike arg max, max may not contain multiple elements: for example, if f is, then, but because the function attains the same value at every element of arg max. Equivalently, if M is the maximum of f, then the arg max 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 the arg max, and arg max is considered a point, not a set of points. So, for example, , since the maximum value of is 25, which occurs for x = 5. However, in case the maximum is reached at many points, arg max needs to be considered a set of points. For example since the maximum value of cos is 1, which occurs on this interval for x = 0, 2π or 4π. On the wholereal line Functions need not in general attain a maximum value, and hence the arg max is sometimes the empty set; for example,, since is unbounded on the realline. As another example,, although arc tan is bounded by ±π/2. However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and thus a nonempty arg max.
Arg min
arg minstands for argument of the minimum, and is defined analogously. For instance, are points x for which f attains its smallest value. It is the complementary operator of.
