Moment Approximations for Set-Semidefinite Polynomials

Article

Abstract

The set of polynomials that are nonnegative over a subset of the nonnegative orthant (we call them set-semidefinite) have many uses in optimization. A common example of this type set is the set of copositive matrices, where we are effectively considering nonnegativity over the entire nonnegative orthant and are restricted to homogeneous polynomials of degree two. Lasserre (SIAM J. Optim., 21(3):864–885, 2011) has previously considered a method using moments in order to provide an outer approximation to this set, for nonnegativity over a general subset of the real space. In this paper, we shall show that, in the special case of considering nonnegativity over a subset of the nonnegative orthant, we can provide a new outer approximation hierarchy. This is based on restricting moment matrices to be completely positive, and it is at least as good as Lasserre’s method. This can then be relaxed to give tractable approximations that are still at least as good as Lasserre’s method. In doing this, we also provide interesting new insights into the use of moments in constructing these approximations.

Keywords

Nonnegative polynomials Set-semidefinite polynomials Copositive programming Doubly nonnegative matrices Moments Completely positive matrices 

Notes

Acknowledgements

All figures in this article were produced using Wolfram Mathematica 8. The authors would like to thank the anonymous referees for their helpful comments.

References

  1. 1.
    Bomze, I.M.: Copositive optimization—Recent developments and applications. Eur. J. Oper. Res. 216(3), 509–520 (2012) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bomze, I.M., Dür, M., Teo, C.-P.: Copositive optimization. Optima Newsl. 89, 2–8 (2012) Google Scholar
  3. 3.
    Burer, S.: Copositive Programming. International Series in Operations Research & Management Science, vol. 166, pp. 201–218. Springer, Berlin (2012) Google Scholar
  4. 4.
    Dür, M.: Copositive programming—a survey. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds.) Recent Advances in Optimization and its Applications in Engineering, pp. 3–20. Springer, Berlin (2010) CrossRefGoogle Scholar
  5. 5.
    Hiriart-Urruty, J.-B., Seeger, A.: A variational approach to copositive matrices. SIAM Rev. 52(4), 593–629 (2010) MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39(2), 117–129 (1987) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120(2), 479–495 (2009) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    de Klerk, E., Pasechnik, D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12(4), 875–892 (2002) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Povh, J., Rendl, F.: A copositive programming approach to graph partitioning. SIAM J. Optim. 18(1), 223–241 (2007) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Povh, J., Rendl, F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discrete Optim. 6(3), 231–241 (2009) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Eichfelder, G., Jahn, J.: Set-semidefinite optimization. J. Convex Anal. 15(4), 767–801 (2008) MathSciNetMATHGoogle Scholar
  12. 12.
    Eichfelder, G., Jahn, J.: Foundations of set-semidefinite optimization. In: Nonlinear Analysis and Variational Problems. Springer Optimization and Its Applications, vol. 35, pp. 259–284. Springer, New York (2010) CrossRefGoogle Scholar
  13. 13.
    Eichfelder, G., Povh, J.: On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets. Optim. Lett. (2013, in print). doi: 10.1007/s11590-012-0450-3
  14. 14.
    Dickinson, P.J.C., Eichfelder, G., Povh, J.: Erratum to the paper “On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets”. Optim. Lett. (2013, to appear) Google Scholar
  15. 15.
    Anjos, M.F., Lasserre, J.B.: Handbook of Semidefinite, Conic and Polynomial Optimization: Theory, Algorithms, Software and Applications. International Series in Operational Research and Management Science, vol. 166. Springer, Berlin (2012) CrossRefGoogle Scholar
  16. 16.
    Lasserre, J.B.: New approximations for the cone of copositive matrices and its dual. Preprint. (2012, submitted). Available at: http://arxiv.org/abs/1012.2552
  17. 17.
    Lasserre, J.B.: A new look at nonnegativity on closed sets and polynomial optimization. SIAM J. Optim. 21(3), 864–885 (2011) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lasserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press Optimization Series, vol. 1. Imperial College Press, London (2010) MATHGoogle Scholar
  19. 19.
    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Emerging Applications of Algebraic Geometry. IMA Vol. Math. Appl., vol. 149, pp. 157–270. Springer, New York (2009) CrossRefGoogle Scholar
  20. 20.
    Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific, River Edge (2003) CrossRefMATHGoogle Scholar
  21. 21.
    Dickinson, P.J.C., Gijben, L.: On the computational complexity of membership problems for the completely positive cone and its dual. Preprint. (2012, submitted). Available at: http://www.optimization-online.org/DB_HTML/2011/05/3041.html
  22. 22.
    Maxfield, J.E., Minc, H.: On the matrix equation XX=A. Proc. Edinb. Math. Soc. 13(02), 125–129 (1962) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Johann Bernoulli InstituteUniversity of GroningenGroningenThe Netherlands
  2. 2.Faculty of information studies in Novo mestoUlica talcev 3Novo mestoSlovenia

Personalised recommendations