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

TL;DR

This paper establishes NP-hardness and approximation bounds for $L_p$-ball constrained homogeneous polynomial optimization problems, extending previous results and providing new algorithms with improved guarantees.

Contribution

It proves NP-hardness for degree-$d$ polynomial optimization over $L_p$-balls for all $p o \infty$, and introduces new approximation algorithms with better guarantees.

Findings

NP-hardness for degree-$d$ polynomial optimization over $L_p$-balls for all $p o \infty$

Deterministic approximation within $igOmega(( ext{log} n)^{(d-2)/p} / n^{d/2-1})$

Randomized approximation within $igOmega(( ext{log} n / n)^{d/2-1})$, independent of $p$

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