Textbook
The function $f:S\to\mathbb R$ has a nonempty domain $S$. A number $m$ is the global minimum on $S$ if $f(x)\ge m$ for every $x\in S$ and some $x_0\in S$ satisfies $f(x_0)=m$. The notation is $m=\min_{x\in S}f(x)$.
Two separate conditions are required: no admissible value is smaller, and at least one equals $m$. For $f(x)=(x-2)^2$ on $\mathbb R$, the square is nonnegative and is zero at 2. Thus the minimum is 0, while 2 is the input attaining it.
The admissible domain is part of the problem. On $S=\{0,1\}$, the same function has values 4 and 1, so its minimum becomes 1. Changing constraints can change the answer.
A bounded-below function need not have a minimum. For $f(x)=x$ on $(0,1)$, every input can be replaced by $x/2$ to get a smaller value; zero is inadmissible. Calling a task minimization does not establish existence of a solution.
The minimum value is unique: if $m_1$ and $m_2$ are both attained lower bounds, then $m_1\le m_2$ and $m_2\le m_1$, hence $m_1=m_2$. Several inputs may nevertheless attain that value. Comparing only nearby inputs does not prove global minimality. The definition compares against every permitted input, including those far from the chosen point.
Definitions and notation: Basic Analysis I; Mathematics for Machine Learning.
Related concepts: Infimum: the greatest lower bound, Argmin: the set of minimizing inputs.
In plain language
A minimum resembles the lowest available price among allowed options. It gives the price rather than the address of a shop charging it. That value must actually occur; an advertised price outside the allowed list cannot supply the purchase. Changing the allowed options can change the minimum even when every price tag stays the same.
Example
The example considers the constructed function $(x-2)^2$, first on $\{0,1,2\}$ and then on $\{0,1\}$.
A function’s minimum: the smallest attained value
Values
Three inputs give values 4, 1, and 0.
Step 1 of 3: Values
Read all steps
Values
Three inputs give values 4, 1, and 0.
- Inputs
- 012
- Values
- 4current1current0current
Attained minimum
Zero is no larger than other values and is attained at input 2.
- Values
- 410included
New constraint
Removing input 2 makes 1 the minimum value.
- Inputs
- 01included2excluded
- Values
- 41included0excluded
In the first set, zero belongs to the value list and is no greater than 4 or 1. Removing input 2 makes zero unattained, although it remains a lower bound for the surviving values. It therefore stops being the new problem’s minimum. Value 1 passes both checks: input 1 attains it, and it is no greater than the other permitted value 4. The constraint changed while the function’s formula stayed fixed.
A minimum requires both conditions: a lower bound and attainment at an admissible input.