Optimal Value: Maximum and Minimum
Simple Definitions
Maximum and Minimum collectively are called the Optimum.
- In the set $X$, the largest element is denoted as the maximum $\max X$, and the smallest element as the minimum $\min X$.
- For the function $f : X \to \mathbb{R}$, the largest function value is denoted as $\max_{X} f$, and the smallest function value as $\min_{X} f$.
- $\mathbb{R}$ denotes the entire set of real numbers.
- Maximum and Minimum are Sino-Korean words, and value is a native Korean word. As of 2021, the correct terms are actually 최댓값 for maximum and 최솟값 for minimum, but these are not very search engine friendly, so the ‘ㅅ’ is omitted. Similarly, using 値 (value) character to write 최대치, 최소치 is also fine, but the most familiar terms have been chosen based on current language habits.
Explanation
From mathematicians’ perspective, whether searching for something large or small does not really matter. Especially in optimization problems, when explaining algorithms in general, it’s simply referred to as optimization, without distinguishing between maximizing or minimizing.
The plural form of Maximum is Maxima, Minimum is Minima, and Optimum is Optima. It does not make sense to have multiple optimum values, hence these expressions should be understood as optimum points, not values, in context.
Optimum value, in contrast with the extreme value, can be considered in the context of optimization problems as the global optimum, whereas the extreme value can be regarded as the local optimum. Usually, the term global is insignificant in this context.
When the value and the set itself are not important in context, the expressions of set, element, and function can be arbitrary, so one must pay close attention.
The significance of this post is actually in defining the familiar terms of maximum and minimum in an unambiguous form of functions. If one can intuitively agree with the simple definition above, let’s also look at the complex definitions below.
Complex Definitions
Optimum of a Set
Let us assume a totally ordered set $\left( Y, \le \right)$ is given.
$\max, \min : 2^{Y} \to Y$ is a function that corresponds to the smallest or the largest element $y_{\ast} \in B$ of each subset $A \in 2^{Y}$ of $Y$. $$ \max : B \mapsto y_{\ast} \ge b \qquad , \forall b \in B \\ \min : B \mapsto y_{\ast} \le b \qquad , \forall b \in B $$
Optimum of a Function
Given a set $X$ as the domain and $Y$ as the codomain of a set of functions $Y^{X}$.
For $A \subset X$, $\max_{A}, \min_{A} : Y^{X} \to Y$ is defined for the function $f : X \to Y$, that is, $f \in Y^{X}$, as follows. $$ \max_{A} f = \max f(A) \\ \min_{A} f = \min f(A) $$
- $f(A)$ is defined as the image of $f$ with respect to $A$, as follows. $$ f(A) := \left\{ f(a) : \forall a \in A \right\} $$
Examples
For the set $[0,1)$, there is no maximum value, and the minimum value is $\min [0, 1) = 0$.
The minimum value of the quadratic function $f(x) := 2x^{2} + 1$ is $\min_{\mathbb{R}} f = f(0) = 1$. If considering a maximum value within $A = [2,3] \subset \mathbb{R}$, it is $\max_{a \in A} f(a) = f(2) = 9$.
Even in the examples above, notations can be messy, but as explained, it’s usually fine to gloss over.