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Generalized Truncated Moment Problems with Unbounded Sets

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Abstract

This paper studies generalized truncated moment problems with unbounded sets. First, we study geometric properties of the truncated moment cone and its dual cone of nonnegative polynomials. By the technique of homogenization, we give a convergent hierarchy of Moment-SOS relaxations for approximating these cones. With them, we give a Moment-SOS method for solving generalized truncated moment problems with unbounded sets. Finitely atomic representing measures, or certificates for their nonexistence, can be obtained by the proposed method. Numerical experiments and applications are also given.

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References

  1. Bertsekas, D.: Convex Optimization Theory. Athena Scientific, Belmont (2009)

    MATH  Google Scholar 

  2. Blekherman, G., Fialkow, L.: The core variety and representing measures in the truncated moment problem. J. Oper. Theory 84, 185–209 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Curto, R., Fialkow, L.: Solution of the singular quartic moment problem. J. Oper. Theory 48, 315–354 (2002)

    MathSciNet  MATH  Google Scholar 

  4. Curto, R., Fialkow, L.: Solution of the truncated hyperbolic moment problem. Integral Equ. Oper. Theory 52, 181–218 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Curto, R., Fialkow, L.: Truncated K-moment problems in several variables. J. Oper. Theory 54, 189–226 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Curto, R., Fialkow, L.: An analogue of the Riesz–Haviland theorem for the truncated moment problem. J. Funct. Anal. 225, 2709–2731 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Curto, R., Fialkow, L.: Recursively determined representing measures for bivariate truncated moment sequences. J. Oper. Theory 70, 401–436 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. De Klerk, E., Laurent, M.: A survey of semidefinite programming approaches to the generalized problem of moments and their error analysis. In: Araujo, C., Benkart, G., Praeger, C., Tanbay, B. (eds.) World Women in Mathematics 2018. Association for Women in Mathematics Series, vol. 20. Springer, Cham (2019)

    Google Scholar 

  9. Di Dio, P., Schmudgen, K.: The multidimensional truncated moment problem: atoms, determinacy, and core variety. J. Funct. Anal. 274, 3124–3128 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Easwaran, C., Fialkow, L.: Positive linear functionals without representing measures. Oper. Matrices 5, 425–434 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fialkow, L.: Truncated multivariable moment problems with finite variety. J. Oper. Theory 60, 343–377 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Fialkow, L.: The truncated K-moment problem: a survey. In: Operator Theory: The State of the Art, Theta Series in Advanced Mathematics, vol. 18, pp. 25–51. Theta Foundation, Bucharest (2016)

  13. Fialkow, L.: The core variety of a multisequence in the truncated moment problem. J. Math. Anal. Appl. 456, 946–969 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fialkow, L., Nie, J.: The truncated moment problem via homogenization and flat extensions. J. Funct. Anal. 263(6), 1682–1700 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fialkow, L., Nie, J.: On the closure of positive flat moment matrices. J. Oper. Theory 69, 257–277 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guo, F., Wang, L., Zhou, G.: Minimizing rational functions by exact Jacobian SDP relaxation applicable to finite singularities. J. Global Optim. 58(2), 261–284 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Henrion, D., Korda, M., Lasserre, J.B.: The Moment-SOS Hierarchy. World Scientific, Singapore (2020)

    Book  MATH  Google Scholar 

  18. Henrion, D., Lasserre, J., Loefberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4–5), 761–779 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hillar, C., Nie, J.: An elementary and constructive solution to Hilbert’s 17th problem for matrices. Proc. Am. Math. Soc. 136(1), 73–76 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Huang, L., Nie, J., Yuan, Y.: Homogenization for polynomial optimization with unbounded sets. Math. Program. (2021) (to appear)

  21. Lasserre, J.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lasserre, J.: A semidefinite programming approach to the generalized problem of moments. Math. Program. 112, 65–92 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lasserre, J.: An Introduction to Polynomial and Semi-algebraic Optimization. Cambridge University Press, Cambridge (2015)

    Book  MATH  Google Scholar 

  24. Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry, pp. 157–270. Springer, Berlin (2009)

    Chapter  Google Scholar 

  25. Nie, J.: Discriminants and nonnegative polynomials. J. Symb. Comput. 47(2), 167–191 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nie, J.: Polynomial matrix inequality and semidefinite representation. Math. Oper. Res. 36(3), 398–415 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nie, J.: Sum of squares methods for minimizing polynomial forms over spheres and hypersurfaces. Front. Math. China 7(2), 321–346 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nie, J.: The A-truncated K-moment problem. Found. Comput. Math. 14, 1243–1276 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nie, J.: Optimality conditions and finite convergence of Lasserre’s hierarchy. Math. Program. 146(1–2), 97–121 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Nie, J.: Linear optimization with cones of moments and nonnegative polynomials. Math. Program. 153(1), 247–274 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nie, J.: Symmetric tensor nuclear norms. SIAM J. Appl. Algebra Geom. 1(1), 599–625 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nie, J., Yang, Z., Zhang, X.: A complete semidefinite algorithm for detecting copositive matrices and tensors. SIAM J. Optim. 28(4), 2902–2921 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Nie, J., Zhang, X.: Real eigenvalues of nonsymmetric tensors. Comput. Optim. Appl. 70(1), 1–32 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nie, J., Tang, X., Yang, Z., Zhong, S.: Dehomogenization for completely positive tensors. Preprint, arXiv:2206.12553 (2022)

  35. Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3), 969–984 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sturm, J.F.: SeDuMi 1.02: a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tang, G., Shah, P.: Guaranteed tensor decomposition: a moment approach. In: Proceedings of the 32nd International Conference on Machine Learning, vol. 37, pp. 1491–1500 (2015)

  38. Tchakaloff, V.: Formules de cubatures mécanique à coefficients non négatifs. Bull. Sci. Math. 81(2), 123–134 (1957)

    MathSciNet  MATH  Google Scholar 

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Funding

Lei Huang and Ya-Xiang Yuan are partially supported by the National Natural Science Foundation of China (No. 12288201).

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Authors and Affiliations

  1. Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Lei Huang & Ya-Xiang Yuan

  2. Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, 92093, USA

    Jiawang Nie

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Lei Huang

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  1. Lei Huang
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  2. Jiawang Nie
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  3. Ya-Xiang Yuan
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Correspondence to Lei Huang.

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Cite this article

Huang, L., Nie, J. & Yuan, YX. Generalized Truncated Moment Problems with Unbounded Sets. J Sci Comput 95, 15 (2023). https://doi.org/10.1007/s10915-023-02139-z

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  • DOI: https://doi.org/10.1007/s10915-023-02139-z

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