Journal of Global Optimization

, Volume 59, Issue 2–3, pp 227–242 | Cite as

Lattice preconditioning for the real relaxation branch-and-bound approach for integer least squares problems

  • Miguel F. AnjosEmail author
  • Xiao-Wen Chang
  • Wen-Yang Ku


The integer least squares problem is an important problem that arises in numerous applications. We propose a real relaxation-based branch-and-bound (RRBB) method for this problem. First, we define a quantity called the distance to integrality, propose it as a measure of the number of nodes in the RRBB enumeration tree, and provide computational evidence that the size of the RRBB tree is proportional to this distance. Since we cannot know the distance to integrality a priori, we prove that the norm of the Moore–Penrose generalized inverse of the matrix of coefficients is a key factor for bounding this distance, and then we propose a preconditioning method to reduce this norm using lattice reduction techniques. We also propose a set of valid box constraints that help accelerate the RRBB method. Our computational results show that the proposed preconditioning significantly reduces the size of the RRBB enumeration tree, that the preconditioning combined with the proposed set of box constraints can significantly reduce the computational time of RRBB, and that the resulting RRBB method can outperform the Schnorr and Eucher method, a widely used method for solving integer least squares problems, on some types of problem data.


Integer least squares Branch-and-bound methods Lattice reduction Preconditioning Box constraints 



We are grateful to two anonymous referees for their detailed criticisms that helped us improve the paper. We also acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada.


  1. 1.
    Aardal, K., Eisenbrand, F.: The LLL algorithm and integer programming. In: Nguyen, P.Q., Vallée, B. (eds.) The LLL Algorithm, Information Security and Cryptography, pp. 293–314. Springer, Berlin (2010)Google Scholar
  2. 2.
    Aardal, K., Heymann, F.: On the structure of reduced kernel lattice bases. In: Goemans, M., Correa, J. (eds.) Integer Programming and Combinatorial Optimization, volume 7801 of Lecture Notes in Computer Science, pp. 1–12. Springer, Berlin (2013)Google Scholar
  3. 3.
    Aardal, K., Weismantel, R., Wolsey, L.A.: Non-standard approaches to integer programming. Discret. Appl. Math. 123, 5–74 (2002)CrossRefGoogle Scholar
  4. 4.
    Aardal, K., Wolsey, L.A.: Lattice based extended formulations for integer linear equality systems. Math. Program. 121, 337–352 (2010)CrossRefGoogle Scholar
  5. 5.
    Agrell, E., Eriksson, T., Vardy, A., Zeger, K.: Closest point search in lattices. IEEE Trans. Inf. Theory 48, 2201–2214 (2002)CrossRefGoogle Scholar
  6. 6.
    Ajtai, M., Dwork, C.: A public-key cryptosystem with worst-case/average-case equivalence. In: Proceedings of STOC, pp. 284–293 (1997)Google Scholar
  7. 7.
    Ajtai, M., Kumar, R., Sivakumar, D.: Sampling short lattice vectors and the closest lattice vector problem. In: Proceedings of CCC, pp. 53–57 (2002)Google Scholar
  8. 8.
    Babai, L.: On Lóvasz’ lattice reduction and the nearest lattice point problem. Combinatorica 6, 1–13 (1986)CrossRefGoogle Scholar
  9. 9.
    Bremmer, M.R.: Lattice Basis Reduction, an Introduction to the LLL Algorithm and Its Applications. CRC Press, Boca Raton (2012)Google Scholar
  10. 10.
    Buchheim, C., Caprara, A., Lodi, A.: An effective branch-and-bound algorithm for convex quadratic integer programming. Math. Program. 135, 369–395 (2012)CrossRefGoogle Scholar
  11. 11.
    Chang, X.-W., Golub, G.H.: Solving ellipsoid-constrained integer least squares problems. SIAM J. Matrix Anal. Appl. 31, 1071–1089 (2009)CrossRefGoogle Scholar
  12. 12.
    Chang, X.-W., Paige, C.C.: Euclidean distances and least squares problems for a given set of vectors. Appl. Numer. Math. 57, 1240–1244 (2007)CrossRefGoogle Scholar
  13. 13.
    Chang, X.-W., Wen, J., Xie, X.: Effects of the LLL reduction on the success probability of the Babai point and on the complexity of sphere decoding. IEEE Trans. Inf. Theory 59, 4915–4926 (2013)CrossRefGoogle Scholar
  14. 14.
    Chang, X.W., Xie, X., Zhou, T.: MILES: MATLAB package for solving Mixed Integer LEast Squares problems, Version 2.0, October 2011.
  15. 15.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)CrossRefGoogle Scholar
  16. 16.
    Eisenbrand, F.: Integer programming and algorithmic geometry of numbers. In: Jünger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.) 50 Years of Integer Programming 1958–2008, pp. 505–559. Springer, Berlin (2010)Google Scholar
  17. 17.
    Fincke, U., Pohst, M.: A procedure for determining algebraic integers of given norm. Proc. RUROCAL 162, 194–202 (1983)Google Scholar
  18. 18.
    Hanrot, G., Pujol, X., Stehle, D.: Algorithms for the shortest and closest lattice vector problems. In: Proceedings of the IWCC, pp. 159–190 (2011)Google Scholar
  19. 19.
    Hassibi, A., Boyd, S.: Integer parameter estimation in linear models with applications to GPS. IEEE Trans. Signal Process. 46, 2938–2952 (1998)CrossRefGoogle Scholar
  20. 20.
  21. 21.
    Kannan, R.: Improved algorithms for integer programming and related lattice problems. In: Proceedings of the STOC, pp. 99–108 (1983)Google Scholar
  22. 22.
    Kisialiou, M., Luo, Z.Q.: Performance analysis of quasi-maximumlikelihood detector based on semi-definite programming. In: Proceedings of the IEEE ICASSP, pp. 433–436 (2005)Google Scholar
  23. 23.
    Krishnamoorthy, B., Pataki, G.: Column basis reduction and decomposable knapsack problems. Discret. Optim. 6, 242–270 (2009)CrossRefGoogle Scholar
  24. 24.
    Ku, W.Y.: Lattice Preconditioning for the Real Relaxation Based Branch and Bound Method for Integer Least Squares Problems. MSc Thesis, School of Computer Science, McGill University (2011)Google Scholar
  25. 25.
    Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515–534 (1982)CrossRefGoogle Scholar
  26. 26.
    Lenstra Jr, H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)CrossRefGoogle Scholar
  27. 27.
    Mehrotra, S., Li, Z.: Branching on hyperplane methods for mixed integer linear and convex programming using adjoint lattices. J. Glob. Optim. 49, 623–649 (2011)CrossRefGoogle Scholar
  28. 28.
    Micciancio, D., Voulgaris, P.: Faster exponential time algorithms for the shortest vector problem. In: Proceedings of SODA, pp. 1468–1480 (2010)Google Scholar
  29. 29.
    Micciancio, D., Voulgaris, P.: A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations. SIAM J. Comput. 42, 1364–1391 (2013)CrossRefGoogle Scholar
  30. 30.
    Pataki, G., Tural, M., Wong, E.B.: Basis reduction and the complexity of branch-and-bound. In: Proceedings of SODA, pp. 1254–1261 (2010)Google Scholar
  31. 31.
    Schnorr, C.P., Euchner, M.: Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math. Program. 66, 181–199 (1994)CrossRefGoogle Scholar
  32. 32.
    Schnorr, C.P.: Progress on lll and lattice reduction. In: Nguyen, P.Q., Vallée, B. (eds.) The LLL Algorithm, Information Security and Cryptography, pp. 145–178. Springer, Berlin (2010)Google Scholar
  33. 33.
    Tan, P., Rasmussen, L.K.: The application of semidefinite programming for detection in CDMA. IEEE J. Sel. Areas Commun. 19, 1442–1449 (2001)CrossRefGoogle Scholar
  34. 34.
    Teunissen, P.J.G., Kleusberg, A.: GPS for Geodesy. Springer, Berlin (1998)Google Scholar
  35. 35.
    van Emde Boas, P.: Another NP-Complete Partition Problem and the Complexity of Computing Short Vectors in a Lattice. Technical Report Rep. 81–04, Mathematics Institute, Amsterdam, The Netherlands (1981)Google Scholar
  36. 36.
    Xie, X., Chang, X.W., Al Borno, M.: Partial LLL reduction. In: Proceedings of IEEE GLOBECOM, 5 pp (2011)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Miguel F. Anjos
    • 1
    • 2
    Email author
  • Xiao-Wen Chang
    • 3
  • Wen-Yang Ku
    • 4
  1. 1.Canada Research Chair in Discrete Nonlinear Optimization in EngineeringGERADMontrealCanada
  2. 2.École Polytechnique de MontréalMontrealCanada
  3. 3.School of Computer ScienceMcGill UniversityMontrealCanada
  4. 4.Department of Mechanical and Industrial EngineeringUniversity of TorontoTorontoCanada

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