Hardness and Approximation Results for $L_p$-Ball Constrained Homogeneous Polynomial Optimization Problems

In this paper, we establish hardness and approximation results for various $L_p$-ball constrained homogeneous polynomial optimization problems, where $p \in [2,\infty]$\mathrm{.}Specifically,weprovethatforanygiven$d \ge 3$and$p \in [2,\infty]$,boththeproblemofoptimizingadegree-$d$ 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 $\mathrm{\Omega }((\log n{)}^{(d-2)\mathrm{/}p}/n^{d\mathrm{/}2-1})$ 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 $\Omega((\log n/n)^{d/2-1})$,whichisindependentof$p$ 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.