Textbook
For $f:S\to\mathbb R$, the set-valued convention applies: argmin contains every admissible input whose function value is no greater than the value at any other input.
$$\operatorname*{argmin}_{x\in S}f(x):=\{x\in S\mid\forall z\in S:\ f(x)\le f(z)\}.$$
A minimum is an output value; argmin contains inputs. For $(x-2)^2$ on the real line, the minimum is 0, but the minimizer set is $\{2\}$. For $(x^2-1)^2$, the same value 0 is attained twice and argmin is $\{-1,1\}$.
Restricting the domain changes the solution set: imposing $x\ge0$ on the second function leaves $\{1\}$. If a minimum is not attained, argmin is empty. For $f(x)=x$ on $(0,1)$, any input improves when replaced by $x/2$.
A particular chosen solution is written $x^*\in\operatorname*{argmin}_{x\in S}f(x)$. Applied notation with equality may assume uniqueness or a tie-breaking rule. Returning only one element in code requires an explicit selection rule.
The definition makes a global demand over the whole domain $S$. Testing several points alone cannot rule out smaller values between them. For $(x^2-1)^2$, nonnegativity of a square proves completeness: no value is below zero, and zero occurs exactly when $x^2=1$. This establishes both the optimality of the displayed inputs and the absence of any other minimizers on the real line.
Definitions and notation: Mathematics for Machine Learning.
Related concepts: A function’s minimum: the smallest attained value.
In plain language
Argmin resembles a list of the cheapest routes, whereas the minimum gives only their cost. Equal best prices leave all matching routes on the list. Allowed options are fixed beforehand: connections, overnight travel, and ticket availability change the search domain. Selecting one route requires a separate rule; a low fare cannot provide a ride to the station by itself.
Example
For the constructed function $(x^2-1)^2$, the example checks inputs $-1,0,1$, then restricts the domain to nonnegative ones.
Argmin: the set of minimizing inputs
Output values
Inputs −1, 0, and 1 produce 0, 1, and 0.
Step 1 of 4: Output values
Read all steps
Output values
Inputs −1, 0, and 1 produce 0, 1, and 0.
- Inputs
- −101
- Outputs
- 0current10current
Minimum value
Zero is attained at two inputs.
- Outputs
- 0included1excluded0included
Input set
Argmin contains −1 and 1, not the output value 0.
- Minimizers
- −1included1included
Additional constraint
The constraint x ≥ 0 excludes −1 and leaves 1.
- Candidates
- −1excluded1included
The three displayed inputs reveal two solutions, but the function’s structure supplies the proof. Values at −1 and 1 attain the universal lower bound zero, and solving $x^2-1=0$ lists every place where that happens. The constraint $x\ge0$ removes the inadmissible input −1. It neither changes the value at 1 nor turns the minimum output into the result of argmin.
The minimum value, the solution set, and a rule selecting one solution are distinct objects.