Mathematical Programming

, Volume 109, Issue 1, pp 1–26

Semidefinite representations for finite varieties

  • Monique Laurent
Article

Abstract

We consider the problem of minimizing a polynomial over a set defined by polynomial equations and inequalities. When the polynomial equations have a finite set of complex solutions, we can reformulate this problem as a semidefinite programming problem. Our semidefinite representation involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space ℝ[x1, . . . ,xn]/I, where I is the ideal generated by the polynomial equations in the problem. Moreover, we prove the finite convergence of a hierarchy of semidefinite relaxations introduced by Lasserre. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem to optimality.

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References

  1. 1.
    Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Mathematical Programming 58, 295–324 (1993)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer, 2003Google Scholar
  3. 3.
    Burer, S., Monteiro, R.D.C., Zhang, Y.: Rank-two heuristics for max-cut and other binary quadratic programs. SIAM Journal on Optimization 12, 503–521 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Burer, S., Monteiro, R.D.C., Zhang, Y.: Maximum stable set formulations and heuristics based on continuous optimization. Mathematical Programming 94, 137–166 (2002)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Cox, D.A., Little, J.B., O'Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, 1997Google Scholar
  6. 6.
    Cox, D.A., Little, J.B., O'Shea, D.: Using Algebraic Geometry. Graduate Texts in Mathematics, Number 185, Springer, New York, 1998Google Scholar
  7. 7.
    Curto, R.E., Fialkow, L.A.: Solution of the truncated complex moment problem for flat data. Memoirs of the American Mathematical Society vol. 119, n. 568 (1996)Google Scholar
  8. 8.
    Fuglede, B.: The multidimensional moment problem. Expositiones Mathematicae 1, 47–65 (1983)MATHMathSciNetGoogle Scholar
  9. 9.
    Jibetean, D., Laurent, M.: Semidefinite approximations for global unconstrained polynomial optimization. SIAM Journal on Optimization 16, 490–514 (2005)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Landau, H.: Classic background of the moment problem. In Moments in Mathematics, Proceedings of Symposia in Applied Mathematics, vol. 37, AMS, Providence, 1987, pp. 1–15Google Scholar
  11. 11.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization 11, 796–817 (2001)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Lasserre, J.B.: An explicit exact SDP relaxation for nonlinear 0–1 programs. In: K. Aardal, A.M.H. Gerards, (eds.), Lecture Notes in Computer Science 2081, 293–303 (2001)MATHMathSciNetGoogle Scholar
  13. 13.
    Lasserre, J.B.: Polynomials nonnegative on a grid and discrete representations. Transactions of the American Mathematical Society 354, 631–649 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Laurent, M.: A comparison of the Sherali-Adams, Lovász-Schrijver and Lasserre relaxations for 0-1 programming. Mathematics of Operations Research 28, 470–496 (2003)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Laurent, M.: Semidefinite relaxations for Max-Cut. In: The Sharpest Cut: The Impact of Manfred Padberg and His Work. M. Grötschel, ed. MPS-SIAM Series in Optimization 4, 257–290 (2004)MathSciNetGoogle Scholar
  16. 16.
    Laurent, M.: Lower bound for the number of iterations in semidefinite relaxations for the cut polytope. Mathematics of Operations Research 28, 871–883 (2003)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Laurent, M.: Revisiting two theorems of Curto and Fialkow on moment matrices. Preprint, 2004. To appear in Proceedings of the American Mathematical Society 133, 2965–2976 (2005)Google Scholar
  18. 18.
    Lovász, L.: On the Shannon capacity of a graph. IEEE Transactions on Information Theory IT-25, 1–7 (1979)Google Scholar
  19. 19.
    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM Journal on Optimization 1, 166–190 (1991)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Marshall, M.: Optimization of polynomial functions. Canad. Math. Bull. 46, 575–587 (2003)MATHMathSciNetGoogle Scholar
  21. 21.
    Nesterov, Y.: Squared functional systems and optimization problems. In: J.B.G. Frenk, C. Roos, T. Terlaky, S. Zhang, (eds.), High Performance Optimization, Kluwer Academic Publishers, 2000, pp. 405–440Google Scholar
  22. 22.
    Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology, May, 2000Google Scholar
  23. 23.
    Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Mathematical Programming B 96, 293–320 (2003)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Parrilo, P.A.: An explicit construction of distinguished representations of polynomials nonnegative over finite sets. Preprint, ETH, Zürich, 2002Google Scholar
  25. 25.
    Parrilo, P., Sturmfels, B.: Minimizing polynomial functions. In: Algorithmic and quantitative real algebraic geometry, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 60, AMS, 2003, pp. 83–99Google Scholar
  26. 26.
    Powers, V., Wörmann, T.: An algorithm for sums of squares of real polynomials. Journal of Pure and Applied Algebra 127, 99–104 (1998)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal 42, 969–984 (1993)MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Schweighofer, M.: Optimization of polynomials on compact semialgebraic sets. SIAM Journal on Optimization 15, 805–825 (2005)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3, 411–430 (1990)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Shor, N.Z.: An approach to obtaining global extremums in polynomial mathematical programming problems. Kibernetika 5, 102–106 (1987)MATHMathSciNetGoogle Scholar
  31. 31.
    Sturmfels, B.: Solving Systems of Polynomial Equations. CBMS, Regional Conference Series in Mathematics, Number 97, AMS, Providence, 2002Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Monique Laurent
    • 1
  1. 1.CWIAmsterdamThe Netherlands

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