Advertisement

Mathematical Programming

, Volume 146, Issue 1–2, pp 97–121 | Cite as

Optimality conditions and finite convergence of Lasserre’s hierarchy

  • Jiawang NieEmail author
Full Length Paper Series A

Abstract

Lasserre’s hierarchy is a sequence of semidefinite relaxations for solving polynomial optimization problems globally. This paper studies the relationship between optimality conditions in nonlinear programming theory and finite convergence of Lasserre’s hierarchy. Our main results are: (i) Lasserre’s hierarchy has finite convergence when the constraint qualification, strict complementarity and second order sufficiency conditions hold at every global minimizer, under the standard archimedean condition; the proof uses a result of Marshall on boundary hessian conditions. (ii) These optimality conditions are all satisfied at every local minimizer if a finite set of polynomials, which are in the coefficients of input polynomials, do not vanish at the input data (i.e., they hold in a Zariski open set). This implies that, under archimedeanness, Lasserre’s hierarchy has finite convergence generically.

Keywords

Lasserre’s hierarchy Optimality conditions Polynomial optimization Semidefinite program Sum of squares 

Mathematics Subject Classification (2000)

65K05 90C22 90C26 

Notes

Acknowledgments

The author was partially supported by NSF grants DMS-0757212 and DMS-0844775. He would like very much to thank Murray Marshall for communications on the boundary hessian condition.

References

  1. 1.
    Bertsekas, D.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1995)zbMATHGoogle Scholar
  2. 2.
    Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  3. 3.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Third edition. Undergraduate Texts in Mathematics. Springer, New York (1997)Google Scholar
  4. 4.
    Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Graduate Texts in Mathematics, vol. 185. Springer, New York (1998)CrossRefGoogle Scholar
  5. 5.
    Gelfand, I., Kapranov, M., Zelevinsky, A.: Discriminants, resultants, and multidimensional determinants. Mathematics: Theory & Applications, Birkhäuser (1994)Google Scholar
  6. 6.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)CrossRefGoogle Scholar
  7. 7.
    Henrion, D., Lasserre, J., Loefberg, J.: GloptiPoly 3: moments, optimization and semidefinite programming. http://homepages.laas.fr/henrion/software/gloptipoly3
  8. 8.
    Henrion, D., Lasserre, J.B.: GloptiPoly: global optimization over polynomials with Matlab and SeDuMi. ACM Trans. Math. Soft. 29, 165–194 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Henrion, D., Lasserre, J.B.: Detecting global optimality and extracting solutions in GloptiPoly. Positive polynomials in control. Lecture Notes in Control and Information Science, vol. 312, pp. 293–310, Springer, Berlin (2005)Google Scholar
  10. 10.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2009)CrossRefGoogle Scholar
  12. 12.
    Laurent, M.: Semidefinite representations for finite varieties. Math. Program. 109, 1–26 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry, vol. 149, pp. 157–270. Springer, Berlin (2009)CrossRefGoogle Scholar
  14. 14.
    Marshall, M.: Representation of non-negative polynomials with finitely many zeros. Annales de la Faculte des Sciences Toulouse 15, 599–609 (2006)CrossRefzbMATHGoogle Scholar
  15. 15.
    Marshall, M.: Positive Polynomials and Sums of Squares Mathematical Surveys and Monographs, vol. 146. American Mathematical Society, Providence (2008)CrossRefGoogle Scholar
  16. 16.
    Marshall, M.: Representation of non-negative polynomials, degree bounds and applications to optimization. Canad. J. Math. 61, 205–221 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Nie, J., Ranestad, K.: Algebraic degree of polynomial optimization. SIAM J. Optim. 20(1), 485–502 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Nie, J.: Discriminants and nonnegative polynomials. J. Symb. Comput. 47(2), 167–191 (2012)CrossRefzbMATHGoogle Scholar
  19. 19.
    Nie, J.: An exact Jacobian SDP relaxation for polynomial optimization. Math. Program. Ser. A 137, 225–255 (2013)CrossRefzbMATHGoogle Scholar
  20. 20.
    Nie, J.: Certifying convergence of Lasserre’s hierarchy via flat truncation. Mathematical Programming, to appearGoogle Scholar
  21. 21.
    Nie, J.: Polynomial optimization with real varieties. Preprint (2012)Google Scholar
  22. 22.
    Putinar, M.: Positive polynomials on compact semi-algebraic sets. Ind. Univ. Math. J. 42, 969–984 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Reznick, B.: Some Concrete Aspects of Hilbert’s 17th problem. In: Contemp. Math., vol. 253, pp. 251–272. American Mathematical Society, Providence (2000)Google Scholar
  24. 24.
    Scheiderer, C.: Non-existence of degree bounds for weighted sums of squares representations. J. Complex. 21, 823–844 (2005)Google Scholar
  25. 25.
    Scheiderer, C.: Distinguished representations of non-negative polynomials. J. Algebra 289, 558–573 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Scheiderer, C.: Sums of squares of regular functions on real algebraic varieties. Trans. Am. Math. Soc. 352, 1039–1069 (1999)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Scheiderer, C.: Positivity and sums of squares: a guide to recent results. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry, in: IMA Volumes Math. Appl., vol. 149, pp. 271–324. Springer, New York (2009)Google Scholar
  28. 28.
    Sturmfels, B.: Solving Systems of Polynomial Equations. CBMS Regional Conference Series in Mathematics, vol. 97. American Mathematical Society, Providence (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California San DiegoLa JollaUSA

Personalised recommendations