SOS approximations of nonnegative polynomials via simple high degree perturbations

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 εr } that are sums of squares (s.o.s). 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 explicit approximation sequence. Then we show that if the moment problem holds for a basic closed semi-algebraic set K S ⊂ 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 as well as the degree and the size of coefficients of the nonnegative polynomial f , but not on the specific values of its coefficients.

Sums of Squares, Moment Matrices and Optimization Over Polynomials

“…While Berg's result is existential, Lasserre and Netzer [77] have provided an explicit and very simple sum of squares approximation, which we present in Theorem 7.2 below. Previously, Lasserre [71] had given an analogous result for polynomials nonnegative on the whole space R n , presented in Theorem 7.3 below.…”

“…We begin with some elementary bounds from [71,77] on the entries of M t (y). As we now see, when M t (y) 0, all entries y α can be bounded in terms of y 0 and y (2t,0,...,0) , .…”

“…Then, for ǫ ≥ (t − 1) t−1 /t t , the polynomial f +ǫx 2t is nonnegative on R and thus a sum of squares (see [77] for details).…”

“…We refer to [69], [77] for further approximation results by sums of squares for polynomials nonnegative on a basic closed semialgebraic set.…”

Sufficient conditions for a real polynomial to be a sum of squares

“…In Theorem 3, Corollary 4 and 5, it is worth noticing the crucial role played by the constant term and the essential monomials (X α i ), as was already the case in [4,6] for approximating nonnegative polynomials by s.o.s.…”