The Effective Lasserre's Perturbative Positivstellensatz

TL;DR

This paper extends Lasserre's perturbation method for sum-of-squares certificates of nonnegative polynomials, providing explicit bounds on the degree needed for approximate positivity certificates over unbounded domains.

Contribution

It introduces a quantitative bound on the SOS truncation order for polynomial positivity certificates using weighted polynomial tails, advancing polynomial optimization techniques.

Findings

N grows polynomially with 1/ε, specifically as (||p||/ε)^{1/(1-t)}

Provides explicit bounds for SOS certificate degree in polynomial optimization

Uses positivity properties of the Mehler kernel and kernel methods

Abstract

We study sum-of-squares (SOS) certificates for nonnegative polynomials $p$ on $\mathbb{R}^d$ and their implications for polynomial optimization over unbounded domains. Building on Lasserre's perturbation approach, we consider SOS representations of $p$ augmented by weighted polynomial tails of the form $\sum_{n=0}^N (x\cdot x)^n/(n!)^t$ for $0 < t < 1$. Our main result provides an explicit quantitative bound on the truncation order $N$ required to achieve an $\varepsilon$-accurate certificate. Using positivity properties of the Mehler kernel and techniques inspired by polynomial kernel methods, we show that $N$ grows polynomially in $1/\varepsilon$, with rate $N = O((\|p\|/\varepsilon)^{1/(1-t)})$.