Sufficiently Regularized Nonnegative Quartic Polynomials are Sum-of-Squares

TL;DR

This work proves that sufficiently large quartic regularization makes shifted quartically regularized cubic polynomials SoS, enabling polynomial-time SDP-based global optimization for these nonconvex subproblems in high-order tensor methods. It develops explicit bounds on the regularization parameter $\eta$ (denoted as $\sigma$) and analyzes both locally convex and locally nonconvex regimes, connecting SoS representability with global optimality. The paper identifies special polynomial subclasses—such as quadratic-quartic forms and certain T-structured quartics—where global optimality guarantees SoS certificates for all $\sigma$, while contrasting with separable quartic norms (Schnabel-like) where SoS may fail. Through theoretical development and preliminary numerical experiments (MGH test set, Nesterov-Chebyshev, Beale function, sensor localization), it demonstrates practical potential for AR3+SoS methods to yield globally certifiable subproblems, albeit with scalability considerations tied to SDP size. These results extend Hilbert’s SoS classifications to a new subfamily motivated by high-order optimization, offering a principled pathway to integrate SoS certificates into high-order tensor-based optimization frameworks.

Abstract

Hilbert's 17th problem famously established that not all nonnegative polynomials admit a sum-of-squares (SoS) representation. Hilbert also identified a few special classes in which nonnegativity and SoS are equivalent, such as univariate polynomials, quadratic polynomials, and bivariate quartic polynomials. In this paper, we extend this equivalence to several new subclasses of multivariate quartically regularized polynomials and characterize the NP-hardness boundary of these special-structure polynomials. Specifically, we consider the global optimization of multivariate symmetric cubic polynomials regularized by weighted quartic powers of the Euclidean norm. These special-structure polynomials arise as iterative subproblems in high-order tensor methods for nonconvex optimization problems. We consider shifting these polynomials by their global optimum so as to make them nonnegative, and show that for sufficiently large regularization parameters and under mild assumptions, these polynomials admit a sum-of-squares representation. We also identify several structured subclasses of quartically regularized cubic polynomials for which global optimality of the model implies that nonnegativity is certified by a sum-of-squares decomposition for all values of the regularization parameter, including quadratic-quartic polynomials and quartic polynomials containing a special cubic term that can be decomposed as the product of a quadratic norm and a linear form. We provide counterexamples based on quartic separable norms that demonstrate the crucial role of the Euclidean norm in these representations. Finally, we illustrate how these SoS-based certificates can be used for Taylor subproblems arising in high-order tensor methods for nonconvex optimization, with encouraging numerical results.

Figures (6)

• Figure 1: Left: Homogeneous cubic model $g_3(s) + \sigma\|s\|^4$ with random negative coefficients uniformly generated in $[0,-100]$ and zero gradient term. Right: Non-homogeneous model $g(s) = g^Ts+ g_3(s) + \sigma\|s\|^4$, where both the cubic and linear coefficients are randomly generated in $[0,-100]$.
• Figure 2: Performance profile plots for the three methods (ARC, AR3+ARC and AR3-SoS convex).
• Figure 3: Histograms of the maximum regularization parameter $\sigma$ attained for the three methods: AR3 + SoS Convex (left), AR3 + SoS (middle), and AR3 + ARC (right). The count on the vertical axis represents the number of MGH test problems whose maximum $\sigma$ falls within each logarithmic scale bin. For most problems, the max. regularization remains within a numerically stable range ($10^1\sim 10^6$). Only a few outlier problems---notably the badly scaled cases.
• Figure 4: Global convergence behaviours across different initializations. Dark regions indicate convergence to the global minimizer, while lighter regions indicate no convergence after reaching the maximum number of iterations.
• Figure 5: Global convergence behaviours of different initialization for minimizing the Beale function (a) the Newton method, (b) the third-order Newton method, and (c) AR3 + SoS Convex with $\delta=0.01$. Yellow indicates $\|x^{k} - \tilde{x}_{\mathrm{glob}}^*\|/\|\tilde{x}_{\mathrm{glob}}^*\|\leq 10^{-3}$ where $x^{k}$ is the outcome after 350 iterations. Blue indicates all other behaviors, including divergence or convergence to non-optimal points

Theorems & Definitions (68)

• Definition 1.1
• Definition 1.2
• Theorem 1.1
• Lemma 1.1
• Theorem 1.2: Hilbert, hilbert1888darstellung
• Theorem 1.3
• Theorem 1.4
• Example 1.1
• Lemma 2.1
• Lemma 2.2