SOS approximations of nonnegative polynomials via simple high degree perturbations
 Jean B. Lasserre,
 Tim Netzer
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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 \({\{f_{\epsilon r}\}}\) 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 semialgebraic set \({K_{S} \subset \mathbb{R}^n}\) 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 \({\epsilon}\) as well as the degree and the size of coefficients of the nonnegative polynomial f, but not on the specific values of its coefficients.
 Berg C., Christensen J.P.R., Ressel P. (1976) Positive definite functions on Abelian semigroups. Math. Ann. 223, 253–274 CrossRef
 Berg C. (1987) The multidimensional moment problem and semigroups. Proc. Symp. Appl. Math. 37, 110–124
 Blekherman, G.: There are significantly more nonnegative polynomials than sums of squares (preprint)
 Haviland E.K. (1935) On the moment problem for distribution functions in more than one dimension. Am. J. Math. 57, 562–572 CrossRef
 Haviland E.K. (1936) On the moment problem for distribution functions in more than one dimension II. Am. J. Math. 58, 164–168 CrossRef
 Jacobi T., Prestel A. (2001) Distinguished representations of strictly positive polynomials. J. Reine Angew. Math. 532, 223–235
 Kuhlmann S., Marshall M. (2002) Positivity, sums of squares and the multidimensional moment problem. Trans. Am. Math. Soc. 354:4285–4301 CrossRef
 Kuhlmann S., Marshall M., Schwartz N. (2005) Positivity, sums of squares and the multidimensional moment problem II. Adv. Geom. 5, 583–606 CrossRef
 Lasserre J.B. (2001) Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 CrossRef
 Lasserre J.B. (2005) S.O.S. approximation of polynomials nonnegative on a real algebraic set. SIAM J. Optim. 16, 610–628 CrossRef
 Lasserre J.B. (2006) A sum of squares approximation of nonnegative polynomials. SIAM J. Optim. 16, 751–765 CrossRef
 Netzer, T.: High degree perturbations of nonnegative polynomials. Diploma Thesis, Department of Mathematics and Statistics, University of Konstanz, Germany, June 2005
 Nussbaum A.E. (1966) Quasianalytic vectors. Ark. Mat. 6, 179–191
 Parrilo P.A. (2003) Semidefinite programming relaxations for semialgebraic problems. Math. Progr. Ser. B 96, 293–320 CrossRef
 Prestel A., Delzell C.N. (2001) Positive Polynomials. Springer, Berlin Heidelberg New York
 Putinar M. (1993) Positive polynomials on compact semialgebraic sets. Indiana Univ. Math. J. 42:969–984 CrossRef
 Scheiderer C. (2003) Positivity and sums of squares: a guide to some recent results. Department of Mathematics, University of Duisburg, Germany
 Schmüdgen K. (1991) The Kmoment problem for compact semialgebraic sets. Math. Ann. 289, 203–206 CrossRef
 Schweighofer M. (2005) Optimization of polynomials on compact semialgebraic sets. SIAM J. Optim. 15, 805–825 CrossRef
 Vandenberghe L., Boyd S. (1996) Semidefinite programming. SIAM Rev 38, 49–95 CrossRef
 Title
 SOS approximations of nonnegative polynomials via simple high degree perturbations
 Journal

Mathematische Zeitschrift
Volume 256, Issue 1 , pp 99112
 Cover Date
 20070501
 DOI
 10.1007/s0020900600618
 Print ISSN
 00255874
 Online ISSN
 14321823
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Real algebraic geometry
 Positive polynomials
 Sum of squares
 Semidefinite programming
 Moment problem
 12E05
 12Y05
 90C22
 44A60
 Authors

 Jean B. Lasserre ^{(1)}
 Tim Netzer ^{(2)}
 Author Affiliations

 1. LAASCNRS and Institute of Mathematics, LAAS, 7 avenue du Colonel Roche, 31077, Toulouse cedex 4, France
 2. Universität Konstanz, Fachbereich Mathematik und Statistik, 78457, Konstanz, Germany