- Hardness and Approximation Results for $L_p$-Ball Constrained Homogeneous Polynomial Optimization Problems Ke Hou(khouse.cuhk.edu.hk) Anthony Man-Cho So(manchosose.cuhk.edu.hk) Abstract: 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. Keywords: Polynomial Optimization; Approximation Algorithms; Diameters of Convex Bodies; Convex Programming Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization ) Category 2: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: Working paper, Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, October 2012. Download: [PDF]Entry Submitted: 10/31/2012Entry Accepted: 10/31/2012Entry Last Modified: 10/31/2012Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.