Arg max

Si_sinc.svg functions above have $$\operatorname{argmax}$$ of {0} because both attain their global maximum value of 1 at x = 0.

The unnormalised sinc function (red) has arg min of {&minus;4.49, 4.49}, approximately, because it has 2 global minimum values of approximately &minus;0.217 at x = ±4.49. However, the normalised sinc function (blue) has arg min of {&minus;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, $f\colon X \to Y$, the $$\operatorname{argmax}$$ over some subset $$S$$ of $$X$$ is defined by


 * $$\operatorname{argmax}_S f := \underset{x \in S}{\operatorname{arg\,max}}\, f(x) := \{x \in S ~:~ f(s) \leq f(x) \text{ for all } s \in S \}.$$

If $$S = X$$ or $$S$$ is clear from the context, then $$S$$ is often left out, as in $$\underset{x}{\operatorname{arg\,max}}\, f(x) := \{ x ~:~ f(s) \leq f(x) \text{ for all } s \in X \}.$$ In other words, $$\operatorname{argmax}$$ is the set of points $$x$$ for which $$f(x)$$ attains the function's largest value (if it exists). $$\operatorname{Argmax}$$ 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 $$Y = [-\infty,\infty] = \mathbb{R} \cup \{ \pm\infty \}$$ are the extended real numbers. In this case, if $$f$$ is identically equal to $$\infty$$ on $$S$$ then $$\operatorname{argmax}_S f := \varnothing$$ (that is, $$\operatorname{argmax}_S \infty := \varnothing$$) and otherwise $$\operatorname{argmax}_S f$$ is defined as above, where in this case $$\operatorname{argmax}_S f$$ can also be written as:
 * $$\operatorname{argmax}_S f := \left\{ x \in S ~:~ f(x) = \sup {}_S f \right\}$$

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 $$\operatorname{argmin}$$ (or $$\operatorname{arg\,min}$$), which stands for argument of the minimum, is defined analogously. For instance,


 * $$\underset{x \in S}{\operatorname{arg\,min}} \, f(x) := \{ x \in S ~:~ f(s) \geq f(x) \text{ for all } s \in S \}$$

are points $$x$$ for which $$f(x)$$ attains its smallest value. It is the complementary operator of $\operatorname{arg\,max}$.

In the special case where $$Y = [-\infty,\infty] = \R \cup \{ \pm\infty \}$$ are the extended real numbers, if $$f$$ is identically equal to $$-\infty$$ on $$S$$ then $$\operatorname{argmin}_S f := \varnothing$$ (that is, $$\operatorname{argmin}_S -\infty := \varnothing$$) and otherwise $$\operatorname{argmin}_S f$$ is defined as above and moreover, in this case (of $$f$$ not identically equal to $$-\infty$$) it also satisfies:
 * $$\operatorname{argmin}_S f := \left\{ x \in S ~:~ f(x) = \inf {}_S f \right\}.$$

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


 * $$\underset{x}{\operatorname{arg\,max}}\, (1 - |x|) = \{ 0 \}.$$

The $$\operatorname{argmax}$$ 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


 * $$\max_x f(x)$$ is the element in $$\{ f(x) ~:~ f(s) \leq f(x) \text{ for all } s \in S \}.$$

Like $$\operatorname{argmax},$$ max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike $$\operatorname{argmax},$$ $$\operatorname{max}$$ may not contain multiple elements: for example, if $$f(x)$$ is $$4 x^2 - x^4,$$ then $$\underset{x}{\operatorname{arg\,max}}\, \left( 4 x^2 - x^4 \right) = \left\{-\sqrt{2}, \sqrt{2}\right\},$$ but $$\underset{x}{\operatorname{max}}\, \left( 4 x^2 - x^4 \right) = \{ 4 \}$$ because the function attains the same value at every element of $$\operatorname{argmax}.$$

Equivalently, if $$M$$ is the maximum of $$f,$$ then the $$\operatorname{argmax}$$ is the level set of the maximum:


 * $$\underset{x}{\operatorname{arg\,max}} \, f(x) = \{ x ~:~ f(x) = M \} =: f^{-1}(M).$$

We can rearrange to give the simple identity


 * $$f\left(\underset{x}{\operatorname{arg\,max}} \, f(x) \right) = \max_x f(x).$$

If the maximum is reached at a single point then this point is often referred to as $$\operatorname{argmax},$$ and $$\operatorname{argmax}$$ is considered a point, not a set of points. So, for example,


 * $$\underset{x\in\mathbb{R}}{\operatorname{arg\,max}}\, (x(10 - x)) = 5$$

(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, $$\operatorname{argmax}$$ needs to be considered a of points.

For example


 * $$\underset{x \in [0, 4 \pi]}{\operatorname{arg\,max}}\, \cos(x) = \{ 0, 2 \pi, 4 \pi \}$$

because the maximum value of $$\cos x$$ is $$1,$$ which occurs on this interval for $$x = 0, 2 \pi$$ or $$4 \pi.$$ On the whole real line


 * $$\underset{x \in \mathbb{R}}{\operatorname{arg\,max}}\, \cos(x) = \left\{ 2 k \pi ~:~ k \in \mathbb{Z} \right\},$$ so an infinite set.

Functions need not in general attain a maximum value, and hence the $$\operatorname{argmax}$$ is sometimes the empty set; for example, $$\underset{x\in\mathbb{R}}{\operatorname{arg\,max}}\, x^3 = \varnothing,$$ since $$x^3$$ is unbounded on the real line. As another example, $$\underset{x \in \mathbb{R}}{\operatorname{arg\,max}}\, \arctan(x) = \varnothing,$$ 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 $$\operatorname{argmax}.$$