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.