The Effective Lasserre's Perturbative Positivstellensatz
Igor Klep, Victor Magron, Matthias Sch\"otz

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 on and their implications for polynomial optimization over unbounded domains. Building on Lasserre's perturbation approach, we consider SOS representations of augmented by weighted polynomial tails of the form for . Our main result provides an explicit quantitative bound on the truncation order required to achieve an -accurate certificate. Using positivity properties of the Mehler kernel and techniques inspired by polynomial kernel methods, we show that grows polynomially in , with rate .
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Taxonomy
TopicsPolynomial and algebraic computation · Geometry and complex manifolds · Advanced Optimization Algorithms Research
