General Problem
Suppose we have a multivariate polynomial function $f(\mathbf{x})$, and several linear functions $\ell_i(\mathbf{x})$. What is known about the complexity of solving the following optimization problem?
\begin{align*} \text{Maximize} & \;\; f(\mathbf{x}) \\ \text{Subject to: } & \;\, \ell_i(\mathbf{x}) \le 0\text{ for all } i \end{align*}
We can assume that the region determined by the constraints is bounded.
Related, but More Specific, Problem
Suppose we have a bounded polytope (represented as the intersection of a set of linear inequalities). I want to compute the maximum volume of an (axis parallel) hyperrectangle completely contained in the polytope. What is the complexity of solving this problem?
Help on either of these problems is greatly appreciated.