Optimization Letters

, Volume 12, Issue 3, pp 499–518 | Cite as

A semidefinite programming method for integer convex quadratic minimization

Original Paper


We consider the NP-hard problem of minimizing a convex quadratic function over the integer lattice \({\mathbf{Z}}^n\). We present a simple semidefinite programming (SDP) relaxation for obtaining a nontrivial lower bound on the optimal value of the problem. By interpreting the solution to the SDP relaxation probabilistically, we obtain a randomized algorithm for finding good suboptimal solutions, and thus an upper bound on the optimal value. The effectiveness of the method is shown for numerical problem instances of various sizes.


Convex optimization Integer quadratic programming Mixed-integer programming Semidefinite relaxation Branch-and-bound 



We thank three anonymous referees for providing helpful comments and constructive remarks.


  1. 1.
    Agrell, E., Eriksson, T., Vardy, A., Zeger, K.: Closest point search in lattices. IEEE Trans. Inf. Theory 48(8), 2201–2214 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arora, S., Babai, L., Stern, J., Sweedyk, Z.: The hardness of approximate optima in lattices, codes, and systems of linear equations. In: Proceedings of the 34th Annual Symposium on Foundations of Computer Science, pp. 724–733. IEEE Computer Society Press (1993)Google Scholar
  3. 3.
    Bienstock, D.: Eigenvalue techniques for convex objective, nonconvex optimization problems. In: Integer Programming and Combinatorial Optimization, pp. 29–42. Springer (2010)Google Scholar
  4. 4.
    Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory, vol. 15. SIAM, Philadelphia (1994)CrossRefMATHGoogle Scholar
  5. 5.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  6. 6.
    Buchheim, C., Caprara, A., Lodi, A.: An effective branch-and-bound algorithm for convex quadratic integer programming. Math. Program. 135(1–2), 369–395 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Buchheim, C., Hubner, R., Schobel, A.: Ellipsoid bounds for convex quadratic integer programming. SIAM J. Optim. 25(2), 741–769 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Buchheim, C., Wiegele, A.: Semidefinite relaxations for non-convex quadratic mixed-integer programming. Math. Program. 141(1–2), 435–452 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chang, X.W., Yang, X., Zhou, T.: MLAMBDA: a modified LAMBDA method for integer least-squares estimation. J. Geod. 79(9), 552–565 (2005)CrossRefMATHGoogle Scholar
  10. 10.
    Chang, X.W., Zhou, T.: MILES: MATLAB package for solving mixed integer least squares problems. GPS Solut. 11(4), 289–294 (2007)CrossRefGoogle Scholar
  11. 11.
    Dinur, I., Kindler, G., Raz, R., Safra, S.: Approximating CVP to within almost-polynomial factors is NP-hard. Combinatorica 23(2), 205–243 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dong, H.: Relaxing nonconvex quadratic functions by multiple adaptive diagonal perturbations. SIAM J. Optim. 26(3), 1962–1985 (2016)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fincke, U., Pohst, M.: Improved methods for calculating vectors of short length in a lattice, including a complexity analysis. Math. Comput. 44(170), 463–471 (1985)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Grant, M., Boyd, S.: Graph implementations for nonsmooth convex programs. In: V. Blondel, S. Boyd, H. Kimura (eds.) Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences, pp. 95–110. Springer-Verlag Limited (2008). http://stanford.edu/~boyd/graph_dcp.html
  16. 16.
    Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1. http://cvxr.com/cvx (2014)
  17. 17.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, vol. 2. Springer, Heidelberg (2012)MATHGoogle Scholar
  18. 18.
    Hassibi, A., Boyd, S.: Integer parameter estimation in linear models with applications to gps. IEEE Trans. Signal Process. 46(11), 2938–2952 (1998)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Jaldén, J., Ottersten, B.: On the complexity of sphere decoding in digital communications. IEEE Trans. Signal Process. 53(4), 1474–1484 (2005)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Knuth, D.E.: Seminumerical Algorithms, The Art of Computer Programming. Addison-Wesley, Boston (1997)MATHGoogle Scholar
  21. 21.
    Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515–534 (1982)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Luo, Z.Q., Ma, W.K., So, A.M.C., Ye, Y., Zhang, S.: Semidefinite relaxation of quadratic optimization problems. IEEE Signal Process. Mag. 27(3), 20–34 (2010)CrossRefGoogle Scholar
  23. 23.
    MOSEK ApS: the MOSEK optimization toolbox for MATLAB manual, version 7.1 (revision 28). http://docs.mosek.com/7.1/toolbox.pdf
  24. 24.
    Nguyen, P.Q., Stern, J.: The two faces of lattices in cryptology. In: Cryptography and Lattices, pp. 146–180. Springer (2001)Google Scholar
  25. 25.
    Schnorr, C.P., Euchner, M.: Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math. Program. 66(1–3), 181–199 (1994)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Ustun, B., Rudin, C.: Supersparse linear integer models for optimized medical scoring systems. Mach. Learn. 102(3), 349–391 (2016)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38(1), 49–95 (1996)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Stanford UniversityStanfordUSA
  2. 2.Stanford UniversityStanfordUSA

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