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 $\Omega\left( (\log n)^{(d-2)/p} \big/ n^{d/2-1} \right)$ 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.

## Citation

Working paper, Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, October 2012.