  In this paper, we establish hardness and approximation results for various
$L_p$-ball constrained homogeneous polynomial optimization problems, where $p
\in [2,\infty]$. Specifically, we prove that for any given $d \ge 3$ and $p \in
[2,\infty]$, both the problem of optimizing a degree-$d$ homogeneous polynomial
over the $L_p$-ball and the problem of optimizing a degree-$d$ multilinear form
(regardless of its super-symmetry) over $L_p$-balls are NP-hard. On the other
hand, we show that these problems can be approximated to within a factor of
$\Omega((\log n)^{(d-2)/p} \big/ n^{d/2-1})$ in deterministic polynomial time,
where $n$ is the number of variables. We further show that with the help of
randomization, the approximation guarantee can be improved to $\Omega((\log
n/n)^{d/2-1})$, which is independent of $p$ and is currently the best for the
aforementioned problems. Our results unify and generalize those in the
literature, which focus either on the quadratic case or the case where $p \in
{2,\infty}$. We believe that the wide array of tools used in this paper will
have further applications in the study of polynomial optimization problems.
