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

In this paper, we establish hardness and approximation results for various Lp–ball constrained homogeneous polynomial optimization problems, where p ∈ [2,∞]. Specifically, we prove that for any given d ≥ 3 and p ∈ [2,∞], both the problem of optimizing a degree–d homogeneous polynomial over the Lp–ball and the problem of optimizing a degree–d multilinear form (regardless of its super–symmetry) over Lp– balls are NP–hard. On the other hand, we show that these problems can be approximated to within a factor of Ω (