Mathematical Programming

, Volume 136, Issue 2, pp 233–251 | Cite as

On convex relaxations for quadratically constrained quadratic programming

Full Length Paper Series B


We consider convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and (possibly nonconvex) quadratic constraints. Let \(\mathcal{F }\) denote the feasible region for the linear constraints. We first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on \(\mathcal{F }\) is dominated by an alternative methodology based on convexifying the range of the quadratic form \(\genfrac(){0.0pt}{}{1}{x}\genfrac(){0.0pt}{}{1}{x}^T\) for \(x\in \mathcal{F }\). We next show that the use of “\(\alpha \)BB” underestimators as computable estimates of convex lower envelopes is dominated by a relaxation of the convex hull of the quadratic form that imposes semidefiniteness and linear constraints on diagonal terms. Finally, we show that the use of a large class of D.C. (“difference of convex”) underestimators is dominated by a relaxation that combines semidefiniteness with RLT constraints.


Quadratically constrained quadratic programming Convex envelope Semidefinite programming Reformulation-linearization technique 

Mathematics Subject Classification

90C26 90C22 


  1. 1.
    Androulakis, I.P., Maranas, C.D., Floudas, C.A.: \(\alpha \)BB: A global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7, 337–363 (1995)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Anstreicher, K.M.: Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming. J. Glob. Optim. 43, 471–484 (2009)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Anstreicher, K.M., Burer, S.: Computable representations for convex hulls of low-dimensional quadratic forms. Math. Prog. B 124, 33–43 (2010)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bomze, I.M.: Branch-and-bound approaches to standard quadratic optimization problems. J. Glob. Optim. 22, 27–37 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Prog. 120, 479–495 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Burer, S., Letchford, A.N.: On nonconvex quadratic programming with box constraints. SIAM J. Optim. 20, 1073–1089 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Burer, S., Saxena, A. : The MILP road to MIQCP. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, IMA Volumes in Mathematics and its Applications, vol. 154, pp. 373–406. Springer (2011)Google Scholar
  8. 8.
    Burer, S., Vandenbussche, D.: A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Math. Prog. 113, 259–282 (2008)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 2nd edn. Springer, Berlin (1993)Google Scholar
  10. 10.
    Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Prog. 103, 251–282 (2005)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Markót, M.C., Csendes, T.: A new verified optimization technique for the “packing circles in a unit square” problems. SIAM J. Optim. 16, 193–219 (2005)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Padberg, M.: The Boolean quadric polytope: some characteristics, facets and relatives. Math. Prog. B 45, 139–172 (1989)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Poljak, S., Wolkowicz, H.: Convex relaxations of (0,1) quadratic programming. Math. Oper. Res. 20, 550–561 (1995)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)MATHGoogle Scholar
  15. 15.
    Sahinidis, N.V.: BARON: a general purpose global optimization software package. J. Glob Optim. 8, 201–205 (1996)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer, Dordrecht (1998)Google Scholar
  17. 17.
    Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Szabó, P.G., Markót, M.C., Csendes, T.: Global optimization in geometry—circle packing into the square. In: Audet, C., Hansen, P., Savard, G. (eds.) Essays and Surveys in Global Optimization, pp. 233–266. Kluwer, Dordrecht (2005)Google Scholar
  19. 19.
    Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38, 49–95 (1996)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Vandenbussche, D., Nemhauser, G.: A branch-and-cut algorithm for nonconvex quadratic programming with box constraints. Math. Prog. 102, 559–575 (2005)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Yajima, Y., Fujie, T.: A polyhedral approach for nonconvex quadratic programming problems with box constraints. J. Glob. Optim. 13, 151–170 (1998)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Zheng, X.J., Sun, X.L., Li, D.: Nonconvex quadratically constrained quadratic programming: best D.C. decompositions and their SDP representations. J. Glob. Optim. 50, 695–712 (2011)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.Department of Management SciencesUniversity of IowaIowa CityUSA

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