A Semidefinite Relaxation Scheme for Multivariate Quartic Polynomial Optimization with Quadratic Constraints

• Authors: Zhi-Quan Luo;Shuzhong Zhang
• Venue: SIAM Journal on Optimization
• Year: 2010

We present a general semidefinite relaxation scheme for general $n$-variate quartic polynomial optimization under homogeneous quadratic constraints. Unlike the existing sum-of-squares approach which relaxes the quartic optimization problems to a sequence of (typically large) linear semidefinite programs (SDP), our relaxation scheme leads to a (possibly nonconvex) quadratic optimization problem with linear constraints over the semidefinite matrix cone in $\mathbb{R}^{n\times n}$. It is shown that each $\alpha$-factor approximate solution of the relaxed quadratic SDP can be used to generate in randomized polynomial time an $O(\alpha)$-factor approximate solution for the original quartic optimization problem, where the constant in $O(\cdot)$ depends only on problem dimension. In the case where only one positive definite quadratic constraint is present in the quartic optimization problem, we present a randomized polynomial time approximation algorithm which can provide a guaranteed relative approximation ratio of $(1-O(n^{-2}))$.