Mathematical Programming

, Volume 166, Issue 1–2, pp 159–184 | Cite as

A fresh CP look at mixed-binary QPs: new formulations and relaxations

  • Immanuel M. Bomze
  • Jianqiang Cheng
  • Peter J. C. Dickinson
  • Abdel Lisser
Full Length Paper Series A


Triggered by Burer’s seminal characterization from 2009, many copositive reformulations of mixed-binary QPs have been discussed by now. Most of them can be used as proper relaxations, if the intractable co(mpletely)positive cones are replaced by tractable approximations. While the widely used approximation hierarchies have the disadvantage to use positive-semidefinite (psd) matrices of orders which rapidly increase with the level of approximation, alternatives focus on the problem of keeping psd matrix orders small, with the aim to avoid memory problems in the interior point algorithms. This work continues this approach, proposing new reformulations and relaxations. Moreover, we provide a thorough comparison of the respective duals and establish a monotonicity relation among their duality gaps. We also identify sufficient conditions for strong duality/zero duality gap in some of these formulations and generalize some of our observations to general conic problems.


Copositivity Completely positive Conic optimization Quadratic optimization Reformulations Nonlinear optimization Nonconvex optimization 

Mathematics Subject Classification

90C20 90C26 90C30 



The authors would like to thank the anonymous handling Associate Editor and two anonymous referees for their very useful comments and suggestions which helped to improve our paper significantly. This research benefited from the support of the “FMJH Program Gaspard Monge in Optimization and Operations Research”, and from the support to this program by EDF.


  1. 1.
    Arima, N., Kim, S., Kojima, M.: Simplified copositive and Lagrangian relaxations for linearly constrained quadratic optimization problems in continuous and binary variables. Pac. J. Optim. 10, 437–451 (2014)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Berman, A.: Cones, matrices and mathematical programming. In: Lecture Notes in Economics and Mathematical Systems. Vol. 79. Springer Verlag (1973)Google Scholar
  3. 3.
    Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific Publication, River Edge, NJ (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bomze, I.M.: Copositive optimization—recent developments and applications. Eur. J. Oper. Res. 216, 509–520 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bomze, I.M.: Copositive relaxation beats Lagrangian dual bounds in quadratically and linearly constrained QPs. SIAM J. Optim. 25, 1249–1275 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bomze, I.M., Jarre, F.: A note on Burers copositive representation of mixed-binary QPs. Optim. Lett. 4, 465–472 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bomze, I.M., Schachinger, W., Ullrich, R.: New lower bounds and asymptotics for the cp-rank. SIAM J. Matrix Anal. Appl. 36(1), 20–37 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bomze, I.M., Schachinger, W., Uchida, G.: Think co(mpletely)positive! Matrix properties, examples and a clustered bibliography on copositive optimization. J. Glob. Optim. 52, 423–445 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120(2), 479–495 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Burer, S.: Optimizing a polyhedral-semidefinite relaxation of completely positive programs. Math. Program. Comput. 2(1), 1–19 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Burer, S.: Copositive programming. In: Anjos, M.F., Lasserre, J.B. (eds.) Handbook of Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications. International Series in Operations Research and Management Science, pp. 201–218. Springer, New York (2012)CrossRefGoogle Scholar
  13. 13.
    Burer, S.: A gentle, geometric introduction to copositive optimization. Math. Program. 151, 89–116 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dickinson, P.J.C.: The copositive cone, the completely positive cone and their generalisations. Ph.D thesis, University of Groningen (2013)Google Scholar
  15. 15.
    Dickinson, P.J.C., Gijben, L.: On the computational complexity of membership problems for the completely positive cone and its dual. Comput. Optim. Appl. 57(2), 403–415 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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
  17. 17.
    Gilbert, J.R., Heath, M.T.: Computing a sparse basis for the null space. SIAM J. Algebraic Discrete Methods 8(3), 446–459 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39(2), 117–129 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pataki, G.: The geometry of semidefinite programming. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.) Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, pp. 29–65. Springer, Berlin (2000)CrossRefGoogle Scholar
  20. 20.
    Pataki, G.: On the closedness of the linear image of a closed convex cone. Math. Oper. Res. 32(2), 395–412 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shaked-Monderer, N., Berman, A., Bomze, I.M., Jarre, F., Schachinger, W.: New results on the cp rank and related properties of co(mpletely )positive matrices. Linear Multilinear Algebra 63(2), 384–396 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shor, N.Z.: Quadratic optimization problems. Izv. Akad. Nauk SSSR Tekhn. Kibernet. 222(1), 128–139 (1987)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Trefethen, L.N., Bau III, D.: Numerical Linear Algebra, vol. 50. SIAM, Philadelphia (1997)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Institut Für Statistik Und Operations Research (ISOR)Universität WienViennaAustria
  2. 2.Department of Systems and Industrial EngineeringUniversity of ArizonaTucsonUSA
  3. 3.Department of Applied MathematicsUniversity of TwenteEnschedeThe Netherlands
  4. 4.Laboratoire de Recherche En Informatique (LRI)Université Paris Sud - XIOrsay CedexFrance

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