Optimal Solution: Maximum and Minimum Factors
Difficult Definitions
Let’s assume an arbitrary set $X$ and a totally ordered set $\left( Y, \le \right)$ are given.
For a subset $S \subset X$ of $X$, the argument of maxima $\argmax_{S} : Y^{X} \to 2^{X}$ and argument of minima $\argmin_{S} : Y^{X} \to 2^{X}$ of the function $f : X \to Y$ are defined as follows. $$ \argmax_{S} f := \left\{ x_{\ast} \in S : f \left( x_{\ast} \right) \ge f(x) , \forall x \in X \right\} \\ \argmin_{S} f := \left\{ x_{\ast} \in S : f \left( x_{\ast} \right) \le f(x) , \forall x \in X \right\} $$
- $2^{X}$ is the power set of $X$, $Y^{X}$ is the set of functions whose domain is $X$ and codomain is $Y$.
- The term ‘argument of maxima’ is not official at all, and is arbitrarily coined by the author due to the lack of a proper conventional term.
Explanation
If you’re studying statistics, you’ll probably first come across this in the method of maximum likelihood estimation.
The argument of maxima and minima are collectively referred to as the optimizer. This is because in an optimization problem where you find the function’s maximum or minimum value, the argument of maxima and minima make up the solution set.
At first, the function seems intimidating because it uses up to six letters of the alphabet. The above definitions are written in the most general and difficult terms possible but to put it simply, they are just the points that make the function’s value the largest or smallest.
The definitions introduced are similar to what you would find on Wikipedia, but honestly, there’s no need to define them in such an overly complicated way by mentioning power sets, function sets, totally ordered sets, etc. While it might make the writer’s life easier, it doesn’t help the reader at all. Let’s look at an example to understand this directly.
Simple Example
If we say $f(x) := \left| (x-a)(x-b) \right|$ then $f : \mathbb{R} \to \mathbb{R}$ has the minimum value $f(a) = f(b) = 0$ at $a,b$. Since the inequality $f(a) = f(b) \le f(x)$ holds for all real numbers $x \in \mathbb{R}$, the argument of minima for $f$ can be expressed as follows. $$ \argmin_{\mathbb{R}} f = \left\{ a, b \right\} $$
Why the Codomain is a Power Set
As seen in the example above, while the function’s minimum value may or may not exist, if it does, it is unique. However, if the function is not injective, multiple arguments may correspond to the same value, which is why.