Optimization Letters

, Volume 7, Issue 6, pp 1373–1386 | Cite as

On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets

Original Paper

Abstract

In the paper we prove that any nonconvex quadratic problem over some set \({K\subset \mathbb {R}^n}\) with additional linear and binary constraints can be rewritten as a linear problem over the cone, dual to the cone of K-semidefinite matrices. We show that when K is defined by one quadratic constraint or by one concave quadratic constraint and one linear inequality, then the resulting K-semidefinite problem is actually a semidefinite programming problem. This generalizes results obtained by Sturm and Zhang (Math Oper Res 28:246–267, 2003). Our result also generalizes the well-known completely positive representation result from Burer (Math Program 120:479–495, 2009), which is actually a special instance of our result with \({K=\mathbb{R}^n_{+}}\).

Keywords

Set-positivity Semidefinite programming Copositive programming Mixed integer programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anstreicher K.M., Burer S.: Computable representations for convex hulls of low dimensional quadratic forms. Math. Program. 124, 33–43 (2010)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bomze I., 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)MATHCrossRefGoogle Scholar
  3. 3.
    Bomze, I., Eichfelder, G.: Copositivity detection by difference-of-convex decomposition and omega-subdivision, Preprint Series of the Institute of Applied Mathematics, University of Erlangen-Nuremberg, vol. 333 (2010)Google Scholar
  4. 4.
    Bomze I., Jarre F.: A note on Burer’s copositive representation of mixed-binary QPs. Optim. Lett. 4, 465–472 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Burer S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120, 479–495 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Burer, S.: Copositive Programming. In: Anjos, M., Lasserre, J.B. (eds.) Handbook of Semidefinite, Conic and Polynomial Optimization, Springer (2011)Google Scholar
  7. 7.
    Burer S., Anstreicher K.M., Dür M.: The difference between 5 × 5 doubly nonnegative and completely positive matrices. Linear Algebra Appl. 431, 1539–1552 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Burer, S., Dong, H.: Representing quadratically constrained quadratic programs as generalized copositive programs. http://www.optimization-online.org/DB_HTML/2011/07/3095.html (2011)
  9. 9.
    Bundfuss S., Dür M.: Algorithmic copositivity detection by simplicial partition. Linear Algebra Appl. 428, 1511–1523 (2008)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Bundfuss S., Dür M.: An adaptive linear approximation algorithm for copositive programs. SIAM J. Optim. 20, 30–53 (2009)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Eichfelder G., Jahn J.: Set-Semidefinite Optimization. J. Convex Anal. 15, 767–801 (2008)MathSciNetMATHGoogle Scholar
  12. 12.
    Eichfelder, G., Jahn, J.: Foundations of Set-Semidefinite Optimization. In: Pardalos, P., Rassias, Th.M., Khan, A.A. (eds.) Nonlinear Analysis and Variational Problems, Chapter 18, pp. 259–284, Springer (2009)Google Scholar
  13. 13.
    Eichfelder G., Povh J.: On the set-semidefinite representation of non-convex quadratic programs with cone constraints. Croat. Oper. Res. Rev. 1, 26–39 (2011)MathSciNetGoogle Scholar
  14. 14.
    Jarre F., Schmallowsky K.: On the computation of C* certificates. J. Glob. Optim. 45, 281–296 (2009)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    de Klerk E., Pasechnik D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12, 875–892 (2002)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Pólik I., Terlaky T.: A survey of the S-lemma. SIAM Rev. 49, 371–418 (2007)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Povh J., Rendl F.: A copositive programming approach to graph partitioning. SIAM J. Optim. 18, 223–241 (2007)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Povh J., Rendl F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discret. Optim. 6, 231–241 (2009)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Sturm J.F., Zhang S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28, 246–267 (2003)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Erlangen-NurembergErlangenGermany
  2. 2.Faculty of information studies in Novo mestoNovo mestoSlovenia

Personalised recommendations