2. Unconstrained Optimization(2th)

2.1 Basic Results on the Existence of Optimizers

2.1. Definition
Let $f:U->\mathbb{R}$ be a function on a set $U\subseteq \mathbb{R}^n$. Let $x^*\in U$ be an arbitrary point, and let $B_r(x^*):=\{x\in U:|| x-x^*||<r\}$ be the open ball of radius $r$ around $x^*$. The point $x^*$ is called
-(a) a local minimizer of $f$ if

$$
f(x^*)\leq f(x)
$$