2. Unconstrained Optimization

2.1 Basic Results on the Existence of Optimizers

2.1.

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
\[
f(x^*)\leq f(x)
\]