What does the following notations mean?
As far as I understood,
Suppose that $f$ is a function which takes $n$ real values and spits out one real value where those input real values are constrained by the set $\Omega$, which itself a proper subset of $\mathbb{R}^n$ (i.e. a set of $n$ real numbers).
A point $x^*$ (different than $\vec x$ and which is a member of set $\Omega$) is a local $minimizer$ of $f$ if there exists a positive value $\varepsilon$ such that:
- $f(x^*)$ is always within $f(x)$ where ???
- What is $\varepsilon$?
Can anyone complete this?
(1.) You can think of $x^*$ as the point in $\Omega$ where $f$ achieves a local minimum.
For example, $f(x)=x^2+1$ achieves a (local) minimum at $0$, where $\Omega=[-1,1]$. The symbol $x^*$ is $0$ in this setting.
(2.) The Greek symbol $\epsilon$ is the radius of the ball $B$ centered at $x^*$ where every point $x$ in $B$ and $\Omega$ satisfies $f(x)\ge f(x^*)$. It is exactly in this sense that $x^*$ is a "local minimizer" of $f$.
Here the ball centered at $y$ of radius $r$, typically denoted $B(y,r)$, is $\{x\in \mathbb{R}^n:\Vert x-y\Vert<r\}$