I've been stuck on a problem and I've not managed to find anything solve it using Google. It goes as follows:
Let $f$ be a (complicated) polynomial on $\mathbb{R}^{n}$, for example $f(x, y, z) = x^3 - yz^5$ (alternatively, any smooth function $f: \mathbb{R}^{n} \to \mathbb{R}$). Now let $D \subset \mathbb{R}^{n}$ be bounded (and closed).
The principal objective is to numerically minimize $f$ on $D$, i.e. find $\min \{f(\textbf{x} \mid \textbf{x} \in D\}$. There are plenty of algorithms to find a local minimum $\textbf{x}_{min}$, such as gradient descent. However, I would like to ensure it is a global minimum as well.
If I could find a lower bound for $f$ on $D$ close (as close as is desired) to $\textbf{x}_{min}$, this would mean $\textbf{x}_{min}$ is indeed a global minimum. However, I have not been successful in finding an algorithm that can calculate such lower bounds. Does anyone know of such an algorithm?
Alternatively, does anyone know of an algorithm applicable here that ensures a global minimum?