Advertisement

On the computation of C* certificates

  • Florian Jarre
  • Katrin Schmallowsky
Article

Abstract

The cone of completely positive matrices C* is the convex hull of all symmetric rank-1-matrices xx T with nonnegative entries. While there exist simple certificates proving that a given matrix \({B\in C^*}\) is completely positive it is a rather difficult problem to find such a certificate. We examine a simple algorithm which—for a given input B—either determines a certificate proving that \({B\in C^*}\) or converges to a matrix \({\bar S}\) in C* which in some sense is “close” to B. Numerical experiments on matrices B of dimension up to 200 conclude the presentation.

Keywords

Completely positive matrices 

References

  1. 1.
    Berman A., Rothblum U.: A note on the computation of the CP-rank. Linear Algebra Appl. 419, 1–7 (2006)CrossRefGoogle Scholar
  2. 2.
    Berman A., Shaked-Monderer N.: Completely Positive Matrices. World Scientific, Singapore (2003)Google Scholar
  3. 3.
    Bomze I.M., Dür M., de Klerk E., Roos C., Quist A.J., Terlaky T.: On copositive programming and standard quadratic optimization problems. J. Glob. Optim. 18, 301–320 (2000)CrossRefGoogle Scholar
  4. 4.
    Bomze, I.M., Jarre, F., Rendl, F.: A quadratic factorization heuristic for copositive programs, Preprint, in preparation (2007)Google Scholar
  5. 5.
    Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Preprint, University of Iowa (2007), available at http://www.optimization-online.org/DB_HTML/2006/10/1501.html
  6. 6.
    de Klerk E., Pasechnik D.V.: A linear programming reformulation of the standard quadratic optimization problem. J. Glob. Optim. 37, 75–84 (2007)CrossRefGoogle Scholar
  7. 7.
    Dür M., Still G.: Interior points of the completely positive cone. Electron. J. Linear Algebra 17, 48–53 (2008) (ISSN 1081-3810)Google Scholar
  8. 8.
    Hall M. Jr., Newman M.: Copositive and completely positive quadratic forms. Proc. Camb. Philos. Soc. 59, 329–339 (1963)CrossRefGoogle Scholar
  9. 9.
    Jarre F., Rendl F.: An augmented primal-dual method for linear conic programs. SIAM J. Optim. 19(2), 808–823 (2008). doi: 10.1137/070687128 CrossRefGoogle Scholar
  10. 10.
    Maxfield J.E., Minc H.: On the matrix equation XX  =  A. Proc. Edinb. Math. Soc. 13, 125–129 (1962/1963)CrossRefGoogle Scholar
  11. 11.
    Murty K.G., Kabadi S.N.: Some NP-complete problems in quadratic and linear programming. Math. Program. 39, 117–129 (1987)CrossRefGoogle Scholar
  12. 12.
    Nesterov Y., Nemirovskii A.: Interior-Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics, Philadelphia (1994)Google Scholar
  13. 13.
    Schmallowsky, K.: On the regularity of second order cone programs and an application to solving large scale problems. Math. Methods Oper. Res. (2008) (to appear)Google Scholar
  14. 14.
    Sturm J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversity of DüsseldorfDüsseldorfGermany

Personalised recommendations