[Paper Review] SOS approximations of nonnegative polynomials via simple high degree perturbation

Jean B. Lasserre, Tim Netzer|ArXiv.org|Oct 21, 2005

Iterative Methods for Nonlinear Equations6 references4 citations

TL;DR

This paper presents a simple, explicit method to approximate any nonnegative polynomial on $[-1,1]^n$ or certain semi-algebraic sets using sums of squares (SOS) via high-degree perturbations. By adding a small multiple of $\varepsilon \Theta_r = \varepsilon(1 + \sum X_j^{2r})$, the resulting polynomial becomes SOS for sufficiently large $r$, enabling effective SOS-based nonnegativity certification and approximation in the $\ell_1$-norm of coefficients.

ABSTRACT

We show that every real polynomial $f$ nonnegative on $[-1,1]^{n}$ can be approximated in the $l_{1}$-norm of coefficients, by a sequence of polynomials $\{f_{\ep r}\}$ that are sums of squares. This complements the existence of s.o.s. approximations in the denseness result of Berg, Christensen and Ressel, as we provide a very simple and extit{explicit} approximation sequence. Then we show that if the Moment Problem holds for a basic closed semi-algebraic set $K_S\subset\R^n$ with nonempty interior, then every polynomial nonnegative on $K_S$ can be approximated in a similar fashion by elements from the corresponding preordering. Finally, we show that the degree of the perturbation in the approximating sequence depends on $ε$ as well as the degree and the size of coefficients of the nonnegative polynomial $f$, but not on the specific values of its coefficients.

Motivation & Objective

• To provide an explicit and constructive SOS approximation sequence for polynomials nonnegative on $[-1,1]^n$, overcoming the non-constructive nature of prior existence results.
• To extend the SOS approximation framework to polynomials nonnegative on basic closed semi-algebraic sets $K_S$ under the Moment Problem assumption.
• To characterize the dependence of the required perturbation degree $r$ on $\varepsilon$, degree, and coefficient size of the polynomial, independent of coefficient values.

Proposed method

• Introduce a perturbation polynomial $\Theta_r = 1 + \sum_{j=1}^n X_j^{2r}$ to construct $f_{\varepsilon r} = f + \varepsilon \Theta_r$, which becomes a sum of squares for large enough $r$.
• Use linear functional techniques and positivity constraints on monomials to bound the values of linear functionals on $\mathcal{A}_{2r}$, ensuring the perturbed polynomial lies in the SOS cone.
• Establish that the minimal $\varepsilon_r^*$ for which $f + \varepsilon_r^* \Theta_r$ is SOS is characterized by a finite-dimensional semidefinite program involving linear forms on $\mathcal{A}_{2r}$.
• Adapt the perturbation method to general semi-algebraic sets $K_S$ by using $\theta_r = \sum_{j=1}^n \sum_{k=0}^r \frac{X_j^{2k}}{k!}$, ensuring $f_{\varepsilon r} \in T_S$ (the preordering) for large $r$.
• Prove that the degree $r$ of the perturbation depends only on $\varepsilon$, $n$, the degree of $f$, and the size of its coefficients—not on the specific coefficient values.
• Leverage induction and moment-based inequalities to bound mixed monomial moments under positivity constraints, ensuring uniform control over the approximation error.

Experimental results

Research questions

• RQ1 Can a constructive, explicit SOS approximation sequence be provided for nonnegative polynomials on $[-1,1]^n$, beyond mere existence?
• RQ2 Does the SOS approximation property extend to polynomials nonnegative on basic closed semi-algebraic sets $K_S$ when the Moment Problem holds for $S$?
• RQ3 What determines the minimal degree $r$ of the perturbation $\Theta_r$ required to make $f + \varepsilon \Theta_r$ a sum of squares?
• RQ4 How does the required perturbation degree $r$ depend on the input polynomial $f$ and the approximation tolerance $\varepsilon$?
• RQ5 Can the perturbation be chosen such that $f_{\varepsilon r}$ lies in the preordering $T_S$, providing a certificate of nonnegativity on $K_S$?

Key findings

• For any polynomial $f$ nonnegative on $[-1,1]^n$, the perturbed polynomial $f_{\varepsilon r} = f + \varepsilon \Theta_r$ is a sum of squares for sufficiently large $r$, with $\|f_{\varepsilon r} - f\|_1 \to 0$ as $\varepsilon \to 0$.
• The minimal $\varepsilon_r^*$ such that $f + \varepsilon_r^* \Theta_r$ is SOS is characterized as the solution to a finite-dimensional semidefinite program involving linear functionals on $\mathcal{A}_{2r}$ with constraints on $L(\Theta_r) \leq 1$ and $L(h^2) \geq 0$ for all $h \in \mathcal{A}_r$.
• For polynomials nonnegative on a basic closed semi-algebraic set $K_S$ with nonempty interior and where the Moment Problem holds, $f_{\varepsilon r} = f + \varepsilon \theta_r$ lies in the preordering $T_S$ for large enough $r$, providing a certificate of nonnegativity.
• The required perturbation degree $r$ depends only on $\varepsilon$, the dimension $n$, the degree of $f$, and the size of its coefficients—not on the specific values of the coefficients of $f$.
• The method avoids deep results like Nussbaum’s theorem and provides a simpler, more direct proof than previous approaches, including those in [7] and [8].
• The approximation is uniform on $[-1,1]^n$, and the perturbation $\theta_r$ ensures that membership in $T_S$ implies nonnegativity on $K_S$, enabling algorithmic detection of nonnegativity.