The $\mathbf {K}$-moment problem for continuous linear functionals
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- by Jean B. Lasserre PDF
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Abstract:
Given a closed (and not necessarily compact) basic semi-algebraic set $\mathbf {K}\subseteq \mathbb {R}^n$, we solve the $\mathbf {K}$-moment problem for continuous linear functionals. Namely, we introduce a weighted $\ell _1$-norm $\ell _{\mathbf {w}}$ on $\mathbb {R}[\mathbf {x}]$, and show that the $\ell _{\mathbf {w}}$-closures of the preordering $P$ and quadratic module $Q$ (associated with the generators of $\mathbf {K}$) is the cone $\textrm {Psd}(\mathbf {K})$ of polynomials nonnegative on $\mathbf {K}$. We also prove that $P$ and $Q$ solve the $\mathbf {K}$-moment problem for $\ell _{\mathbf {w}}$-continuous linear functionals and completely characterize those $\ell _{\mathbf {w}}$-continuous linear functionals nonnegative on $P$ and $Q$ (hence on $\textrm {Psd}(\mathbf {K})$). When $\mathbf {K}$ has a nonempty interior, we also provide in explicit form a canonical $\ell _{\mathbf {w}}$-projection $g^{\mathbf {w}}_f$ for any polynomial $f$, on the (degree-truncated) preordering or quadratic module. Remarkably, the support of $g^{\mathbf {w}}_f$ is very sparse and does not depend on $\mathbf {K}$! This enables us to provide an explicit Positivstellensatz on $\mathbf {K}$. And last but not least, we provide a simple characterization of polynomials nonnegative on $\mathbf {K}$, which is crucial in proving the above results.References
- Robert B. Ash, Real analysis and probability, Probability and Mathematical Statistics, No. 11, Academic Press, New York-London, 1972. MR 0435320
- Christian Berg, Jens Peter Reus Christensen, and Paul Ressel, Positive definite functions on abelian semigroups, Math. Ann. 223 (1976), no. 3, 253–274. MR 420150, DOI 10.1007/BF01360957
- Christian Berg, The multidimensional moment problem and semigroups, Moments in mathematics (San Antonio, Tex., 1987) Proc. Sympos. Appl. Math., vol. 37, Amer. Math. Soc., Providence, RI, 1987, pp. 110–124. MR 921086, DOI 10.1090/psapm/037/921086
- Grigoriy Blekherman, There are significantly more nonnegative polynomials than sums of squares, Israel J. Math. 153 (2006), 355–380. MR 2254649, DOI 10.1007/BF02771790
- J. Cimpric, M. Marshall, T. Netzer, Closures of quadratic modules, Israel J. Math., to appear.
- M. Ghasemi, S. Kuhlmann, E. Samei, The moment problem for continuous positive semidefinite linear functionals, arXiv:1010.279v3, November 2010.
- Didier Henrion, Jean-Bernard Lasserre, and Johan Löfberg, GloptiPoly 3: moments, optimization and semidefinite programming, Optim. Methods Softw. 24 (2009), no. 4-5, 761–779. MR 2554910, DOI 10.1080/10556780802699201
- S. Kuhlmann and M. Marshall, Positivity, sums of squares and the multi-dimensional moment problem, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4285–4301. MR 1926876, DOI 10.1090/S0002-9947-02-03075-1
- S. Kuhlmann, M. Marshall, and N. Schwartz, Positivity, sums of squares and the multi-dimensional moment problem. II, Adv. Geom. 5 (2005), no. 4, 583–606. MR 2174483, DOI 10.1515/advg.2005.5.4.583
- Jean B. Lasserre and Tim Netzer, SOS approximations of nonnegative polynomials via simple high degree perturbations, Math. Z. 256 (2007), no. 1, 99–112. MR 2282261, DOI 10.1007/s00209-006-0061-8
- Jean B. Lasserre, Sufficient conditions for a real polynomial to be a sum of squares, Arch. Math. (Basel) 89 (2007), no. 5, 390–398. MR 2363689, DOI 10.1007/s00013-007-2251-y
- Victoria Powers and Claus Scheiderer, The moment problem for non-compact semialgebraic sets, Adv. Geom. 1 (2001), no. 1, 71–88. MR 1823953, DOI 10.1515/advg.2001.005
- Mihai Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J. 42 (1993), no. 3, 969–984. MR 1254128, DOI 10.1512/iumj.1993.42.42045
- Claus Scheiderer, Positivity and sums of squares: a guide to recent results, Emerging applications of algebraic geometry, IMA Vol. Math. Appl., vol. 149, Springer, New York, 2009, pp. 271–324. MR 2500469, DOI 10.1007/978-0-387-09686-5_{8}
- Konrad Schmüdgen, The $K$-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), no. 2, 203–206. MR 1092173, DOI 10.1007/BF01446568
- K. Schmüdgen, Positive cones in enveloping algebras, Rep. Math. Phys. 14 (1978), no. 3, 385–404. MR 530471, DOI 10.1016/0034-4877(78)90008-3
- Lieven Vandenberghe and Stephen Boyd, Semidefinite programming, SIAM Rev. 38 (1996), no. 1, 49–95. MR 1379041, DOI 10.1137/1038003
Additional Information
- Jean B. Lasserre
- Affiliation: LAAS-CNRS and Institute of Mathematics, University of Toulouse, LAAS, 7 avenue du Colonel Roche, 31077 Toulouse Cédex 4, France
- MR Author ID: 110545
- Email: lasserre@laas.fr
- Received by editor(s): July 19, 2011
- Received by editor(s) in revised form: September 5, 2011, and September 7, 2011
- Published electronically: October 4, 2012
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 2489-2504
- MSC (2010): Primary 44A60, 13B25, 14P10, 30C10
- DOI: https://doi.org/10.1090/S0002-9947-2012-05701-1
- MathSciNet review: 3020106