Notation for the set of all arguments corresponding to local minima.

Question

The notation $$\mathop{\mathrm{arg\, min}}_{x \in X} f(x)$$ is sometimes used for the set of all $x \in X$ corresponding to global minima of the function $x \in X \mapsto f(x).$ Is there notation for the set of all $x \in X$ corresponding to local minima? (I included 'calculus' in the tags because it seems likely that someone within that knowledge base knows the answer.)

Answer 1

Considering only the local minima does not seem particularly natural and it is unlikely that there is a standard piece of notation for it. On the other hand, the set of all extrema can be written simply as the level set of the derivative: $\{f'=0\}$.

Answer 2

Like wrote Liu Gang in the comment, using a compact notation for neighborhood such as for example:

$$ f(\bar{x}) \le f(x) \quad \forall x \in U(\bar{x}) $$

Or also: $$ \bar{x} = \mathop{\mathrm{arg\, min}}_{x \in U(\bar{x})} f(x) $$ where $U(\bar{x}) \in \mathcal{J}(\bar{x})$, a neighborhood of $\bar{x}$.

I never saw more compact notation

Answer 3

I would write as $$\{x\in X:x=\mathop{\arg\min}_{y\in U^\circ,\ U\subset X}f(y)\}$$ where $U^\circ$ is the interior of $U$, hence open.

Answer 4

My goal here is to extend the idea proposed by Liu Gang. You could write $$S:=\{x \in X: x\text{ satisfy property } P\},$$ where $P$ is a property that characterizes local minima. Liu Gang chose $P$ to be the mathematical definition of a local minimum. You could also make it even easier by putting $P$ to be the property "is a local minimum", which would then be

$$S:=\{x \in X: x\text{ is a local minimum of } f\}.$$

Nevertheless you could also use more context. For example if $f: \mathbb{R}\to \mathbb{R}$ is smooth then you could write $$S:=\{x \in \mathbb{R}: \exists n \in \mathbb{N} \text{ with } f'(x) = f''(x)=\ldots = f^{(2n-1)}(x)=0 \text{ and } f^{(2n)}(x) >0\}.$$ Conclusion: through all my readings, I never encountered a symbol that seems to be generally accepted to denote the set of local minima. But it's up to you to propose one and use it in a text that will be THE reference ;).