Skip to main content
Log in

A semidefinite programming approach to the generalized problem of moments

  • FULL LENGTH PAPER
  • Published:
Mathematical Programming Submit manuscript

Abstract

We consider the generalized problem of moments (GPM) from a computational point of view and provide a hierarchy of semidefinite programming relaxations whose sequence of optimal values converges to the optimal value of the GPM. We then investigate in detail various examples of applications in optimization, probability, financial economics and optimal control, which all can be viewed as particular instances of the GPM.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akhiezer N.I. (1965). The Classical Moment Problem. Hafner, New York

    Google Scholar 

  2. Akhiezer, N.I., Krein, M.G.: Some questions in the theory of moments. Am. Math. Soc. Transl. 2, Am. Math. Soc., Providence, R.I. (1962)

  3. Anastassiou, G.A.: Moments in Probability Theory and Approximation Theory, Pitman Research Notes in Mathematics Series. Longman Scientific & Technical (1993)

  4. Ash R. (1972). Real Analysis and Probability. Academic, San Diego

    Google Scholar 

  5. Billingsley P. (1968). Convergence of Probability Measures. Wiley, New York

    MATH  Google Scholar 

  6. Berg, C.: The multidimensional moment problem and semi-groups. In: Landau, H.J. (ed.) Moments in Mathematics. American Mathematical Society, Providence. Proc. Symp. Appl. Math. 37, 110–124 (1980)

  7. Bertsimas D. and Popescu I. (2005). Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15: 780–804

    Article  MATH  MathSciNet  Google Scholar 

  8. Curto R.E. and Fialkow L.A. (1991). Recursiveness, positivity and truncated moment problems. Houston J. Math. 17: 603–635

    MATH  MathSciNet  Google Scholar 

  9. Curto R.E. and Fialkow L.A. (2000). The truncated complex K-moment problem. Trans. Am. Math. Soc. 352: 2825–2855

    Article  MATH  MathSciNet  Google Scholar 

  10. Henrion D. and Lasserre J.B. (2003). GloptiPoly:global optimization over polynomials with matlab and SeDuMi. ACM Trans. Math. Soft. 29: 165–194

    Article  MATH  MathSciNet  Google Scholar 

  11. Hernández-Lerma O. and Lasserre J.B. (2003). Markov Chains and Invariant Probabilities. Birkhäuser Verlag, Basel

    MATH  Google Scholar 

  12. Hernández-Lerma O. and Lasserre J.B. (1998). Approximation schemes for infinite linear programs. SIAM J. Optim. 8: 973–988

    Article  MATH  MathSciNet  Google Scholar 

  13. Isii K. (1960). The extrema of probability determined by generalized moments (I): bounded random variables. Ann. Inst. Math. Stat. 12: 119–133

    Article  MathSciNet  Google Scholar 

  14. Isii K. (1963). On sharpness of Tchebycheff-type inequalities. Ann. Inst. Stat. Math. 14: 185–197

    MATH  MathSciNet  Google Scholar 

  15. Jacobi T. and Prestel A. (2001). Distinguished representations of strictly positive polynomials. J. Reine. Angew. Math. 532: 223–235

    MATH  MathSciNet  Google Scholar 

  16. Jibetean D. and de Klerk E. (2006). Global optimization of rational functions: a semidefinite programming approach. Math. Prog. A 106: 93–109

    Article  MATH  MathSciNet  Google Scholar 

  17. Karlin S. and Studden W.J. (1966). Tchebycheff Systems with Applications in Analysis and Statistics. Interscience, New York

    MATH  Google Scholar 

  18. Kemperman, J.H.B.: Geometry of the moment problem, In: Landau, H.J. (ed.) Moments in Mathematics, American Mathematical Society, Providence. Proc. Symp. Appl. Math. 37, 16–53 (1980)

  19. Krein, M.G., Nudelman, A.A.: Markov Moment Problems and Extremal Problems. Translations of Math. Monographs vol. 50, Am. Math. Soc., Providence, RI (1977)

  20. Landau, H.J.: Moments in mathematics. In: Landau, H.J. (ed.) Proc. Symp. Appl. Math. 37, American Mathematical Society, Providence (1980)

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Lasserre J.B. (2002). An explicit equivalent positive semidefinite program for nonlinear 0-1 programs. SIAM J. Optim. 12: 756–769

    Article  MATH  MathSciNet  Google Scholar 

  23. Lasserre J.B. (2002). Bounds on measures satisfying moment conditions. Ann. Appl. Prob. 12: 1114–1137

    Article  MATH  MathSciNet  Google Scholar 

  24. Lasserre, J.B., Prieur, C., Henrion, D.: Nonlinear optimal control: Numerical approximations via moments and LMI relaxations. In: Proceedings of the 44th IEEE Conference on Decision and Control, Sevilla, December, pp. 1648–1653, Spain (2005)

  25. Lasserre, J.B., Prieur, C., Henrion D., Trélat, E.: Nonlinear optimal control with state constraints: Numerical approximations via moments and LMI relaxations. LAAS-CNRS, Toulouse, France. (in preparation)

  26. Lasserre J.B. and Prieto-Rumeau T. (2004). SDP vs. LP-relaxations for the moment approach in some performance evaluation problems. Stoch. Models 20: 439–456

    MATH  MathSciNet  Google Scholar 

  27. Lasserre J.B., Prieto-Rumeau T. and Zervos M. (2006). Option pricing exotic options via moments and semidefinite relaxations. Math. Financ. 16: 469–494

    Article  MATH  MathSciNet  Google Scholar 

  28. Lasserre J.B. (2006). Convergent semidefinite relaxations in polynomial optimization with sparsity. SIAM J. Optim. 17: 822–843

    Article  MATH  MathSciNet  Google Scholar 

  29. Laurent M. (2005). Revisiting two theorems of Curto and Fialkow on moment matrices. Proc. Amer. Math. Soc. 133: 2965–2976

    Article  MATH  MathSciNet  Google Scholar 

  30. Maserick P.H. and Berg C. (1984). Exponentially bounded positive definite functions. Illinois J. Math. 28: 162–179

    MATH  MathSciNet  Google Scholar 

  31. Nussbaum A.E. (1966). Quasi-analytic vectors. Ark. Mat. 6: 179–191

    Article  MathSciNet  Google Scholar 

  32. Powers V. and Reznick B. (2000). Polynomials that are positive on an interval. Trans. Am. math. Soc. 352: 4677–4692

    Article  MATH  MathSciNet  Google Scholar 

  33. Petersen L.C. (1982). On the relation between the multidimensional moment problem and the one dimensional moment problem. Math. Scand. 51: 361–366

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  35. Schmüdgen K. (1991). The K-moment problem for compact semi-algebraic sets. Math. Ann. 289: 203–206

    Article  MATH  MathSciNet  Google Scholar 

  36. Schweighofer M. (2005). Optimization of polynomials on compact semialgebraic sets. SIAM J. Optim. 15: 805–825

    Article  MATH  MathSciNet  Google Scholar 

  37. Shohat J.A. and Tamarkin J.D. (1943). The Problem of Moments. AMS, New York

    MATH  Google Scholar 

  38. Simon B. (1998). The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137: 82–203

    Article  MATH  MathSciNet  Google Scholar 

  39. Vinter V. (1993). Convex duality and nonlinear optimal control. SIAM J. Control Optim. 31: 518–538

    Article  MATH  MathSciNet  Google Scholar 

  40. Waki S., Kim S., Kojima M. and Maramatsu M. (2006). Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17: 218–242

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jean B. Lasserre.

Additional information

This work was supported by french ANR-grant NT05-3-41612, and part of it was completed in January 2006 while the author was a member of IMS, the Institute for Mathematical Sciences of NUS (The National University of Singapore).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lasserre, J.B. A semidefinite programming approach to the generalized problem of moments. Math. Program. 112, 65–92 (2008). https://doi.org/10.1007/s10107-006-0085-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-006-0085-1

Keywords

Mathematics Subject Classification (2000)

Navigation