Semidefinite Relaxations of Products of Nonnegative Forms on the Sphere

TL;DR

This paper introduces a polynomial-size semidefinite relaxation for maximizing the geometric mean of non-negative quadratic forms on the sphere, providing constant-factor approximation guarantees and extending to higher degrees.

Contribution

It develops a novel, efficient SDP relaxation for a non-convex polynomial optimization problem, with proven approximation bounds and a hierarchy of relaxations for general degrees.

Findings

The relaxation achieves a constant factor approximation for quadratic forms.

The analysis is tight as the degree increases, approaching the approximation bound.

A hierarchy of relaxations and rounding algorithms are proposed for broader applicability.

Abstract

We study the problem of maximizing the geometric mean of $d$ low-degree non-negative forms on the real or complex sphere in $n$ variables. We show that this highly non-convex problem is NP-hard even when the forms are quadratic and is equivalent to optimizing a homogeneous polynomial of degree $O(d)$ on the sphere. The standard Sum-of-Squares based convex relaxation for this polynomial optimization problem requires solving a semidefinite program (SDP) of size $n^{O(d)}$, with multiplicative approximation guarantees of $\Omega(\frac{1}{n})$. We exploit the compact representation of this polynomial to introduce a SDP relaxation of size polynomial in $n$ and $d$, and prove that it achieves a constant factor multiplicative approximation when maximizing the geometric mean of non-negative quadratic forms. We also show that this analysis is asymptotically tight, with a sequence of instances where the gap between the relaxation and true optimum approaches this constant factor as $d \rightarrow \infty$. Next we propose a series of intermediate relaxations of increasing complexity that interpolate to the full Sum-of-Squares relaxation, as well as a rounding algorithm that finds an approximate solution from the solution of any intermediate relaxation. Finally we show that this approach can be generalized for relaxations of products of non-negative forms of any degree.