SOS approximations of nonnegative polynomials via simple high degree perturbations

By Jean B. Lasserre, Tim Netzer. 2005

80 citations recorded by the source corpus.

We show that every real polynomial f nonnegative on [−1,1]n can be approximated in the l1-norm of coefficients, by a sequence of polynomials $${\{f_{\epsilon r}\}}$$ 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 $${K_{S} \subset \mathbb{R}^n}$$ with nonempty interior, then every polynomial nonnegative on KS 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 $${\epsilon}$$ as well as the degree and the size of coefficients of the nonnegative polynomial f, but not on the specific values of its coefficients.

Subfields: Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/668ce33d4d1c5e0b379fa9b11127dc68df1bc24d

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In Computational Theory and Mathematics

1. Monique Laurent
2. Martin J. Wainwright
3. Michael I. Jordan

In Numerical Analysis

1. Monique Laurent
2. Martin J. Wainwright
3. Michael I. Jordan

Citations in the corpus

• Sums of Squares, Moment Matrices and Optimization Over Polynomials (2009, methodology): “[92] Let f ∈ R[x] be a polynomial nonnegative on [−1, 1] and let Θt be as in (7.”