Advertisement

Journal of Global Optimization

, Volume 57, Issue 4, pp 1139–1146 | Cite as

A note on set-semidefinite relaxations of nonconvex quadratic programs

  • Faizan Ahmed
  • Georg Still
Article
  • 201 Downloads

Abstract

We consider semidefinite, copositive, and more general, set-semidefinite programming relaxations of general nonconvex quadratic problems. For the semidefinite case a comparison between the feasible set of the original program and the feasible set of the relaxation has been given by Kojima and Tunçel (SIAM J Optim 10(3):750–778, 2000). In this paper the comparison is presented for set-positive relaxations which contain copositive relaxations as a special case.

Keywords

Quadratic programs Semidefinite- Copositive- and Set-positive relaxations 

Mathematics Subject Classification

90C20 90C22 90C09 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bomze I.M.: Block pivoting and shortcut strategies for detecting copositivity. Linear Algebra Appl. 248, 161–184 (1996)CrossRefGoogle Scholar
  2. 2.
    Bomze I.M., De Klerk E.: Solving standard quadratic optimization problems via linear semidefinite and copositive programming. Dedicated to Professor Naum Z. Shor on his 65th birthday. J. Glob. Optim. 24(2), 163–185 (2002)CrossRefGoogle 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.: A note on Burer’s copositive representation of mixed-binary QPs. Optim. Lett. 4(3), 465–472 (2010)CrossRefGoogle Scholar
  5. 5.
    Bomze I.M., Schachinger W.: Multi-standard quadratic optimization: interior point methods and cone programming reformulation. Comput. Optim. Appl. 45, 237–256 (2010)CrossRefGoogle Scholar
  6. 6.
    Burer S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120, 479–495 (2009)CrossRefGoogle Scholar
  7. 7.
    Burer S., Dong H.: Representing quadratically constrained quadratic programs as generalized copositive programs. Oper. Res. Lett. 40(3), 203–206 (2012)CrossRefGoogle Scholar
  8. 8.
    Burer, S.: Copositive programming. In: Anjos, M., Lasserre, J.B. (eds.) Handbook of Semidefinite, Cone and Polynomial Optimization (to appear) (2010)Google Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Dür M.: Copositive programming—a survey, in Recent Advances. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds) Optimization and its Applications in Engineering, pp. 320. Springer, Berlin (2010)Google Scholar
  11. 11.
    Eichfelder G., Jahn J.: Set-semidefinite optimization. J. Convex Anal. 15(4), 767–801 (2008)Google Scholar
  12. 12.
    Kojima M., Tunçel L.: Cones of matrices and successive convex relaxations of nonconvex sets. SIAM J. Optim. 10(3), 750–778 (2000)CrossRefGoogle Scholar
  13. 13.
    Povh J., Rendl F.: A copositive programming approach to graph partitioning. SIAM J. Optim. 18(1), 223–241 (2007)CrossRefGoogle Scholar
  14. 14.
    Povh J., Rendl F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discret. Optim. 6, 3231241 (2009)CrossRefGoogle Scholar
  15. 15.
    Preisig J.C.: Copositivity and the minimization of quadratic functions with nonnegativity and quadratic equality constraints. SIAM J. Control Optim. 34, 1135–1150 (1996)CrossRefGoogle Scholar
  16. 16.
    Quist A.J., De Klerk E., Roos C., Terlaky T.: Copositive relaxation for general quadratic programming. Optim. Methods Softw. 9, 185–208 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of TwenteEnschedeThe Netherlands

Personalised recommendations