Sufficiently Regularized Nonnegative Quartic Polynomials are Sum-of-Squares

Abstract

A polynomial that is nonnegative need not be a sum of squares of polynomials. This classical gap, identified by Hilbert in 1888, lies at the heart of why the global optimization of multivariate quartic polynomials is NP-hard. Yet we show that this gap is closed when using (sufficient) regularization, which fundamentally alters the algebraic structure of the problem. Namely, we investigate a class of quartically-regularized cubic polynomials which arise naturally in polynomial optimization and higher-order tensor methods for nonconvex problems. We show that, under mild assumptions and for sufficiently large Euclidean quartic regularization, the shifted nonnegative polynomial becomes a sum of squares, yielding an exact semidefinite programming (SDP) formulation at the zeroth level of the Lasserre hierarchy. We further derive explicit bounds on the regularization parameter that guarantee this property. Beyond this asymptotic regime, we identify structured subclasses for which SoS exactness holds for all regularization levels, including quadratic-quartic models and a class of low-rank-type cubic tensors. In contrast, we show that separable quartic regularized polynomials -- including classical tensor models proposed by Schnabel (1991) -- do not, in general, induce SoS representations, even under arbitrarily large regularization. Our results reveal a sharp structural boundary between tractable and intractable regimes in polynomial optimization. In particular, they explain why Euclidean quartic regularization plays a significant role: in addition to regularising the model, it can induce exact SoS certificates and exact SDP representations.