Mathematical Programming

, Volume 135, Issue 1–2, pp 369–395 | Cite as

An effective branch-and-bound algorithm for convex quadratic integer programming

Full Length Paper Series A

Abstract

We present a branch-and-bound algorithm for minimizing a convex quadratic objective function over integer variables subject to convex constraints. In a given node of the enumeration tree, corresponding to the fixing of a subset of the variables, a lower bound is given by the continuous minimum of the restricted objective function. We improve this bound by exploiting the integrality of the variables using suitably-defined lattice-free ellipsoids. Experiments show that our approach is very fast on both unconstrained problems and problems with box constraints. The main reason is that all expensive calculations can be done in a preprocessing phase, while a single node in the enumeration tree can be processed in linear time in the problem dimension.

Keywords

Convex quadratic minimization Closest vector problem Branch-and-bound algorithm Computational results 

Mathematics Subject Classification (2000)

90C10 90C25 90C57 90C90 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    BLAS (Basic Linear Algebra Subprograms) (2009). http://www.netlib.org/blas
  2. 2.
    LAPACK – Linear Algebra PACKage (2009). http://www.netlib.org/lapack
  3. 3.
    Bizzarri, F., Buchheim, C., Callegari, S., Caprara, A., Lodi, A., Rovatti, R., Setti, G.: Practical solution of periodic filtered approximation as a convex quadratic integer program. In: Aiguier, M., Bretaudeau, F., Krob, D. (eds.) Complex Systems Design & Management (CSDM) 2010, pp. 149–160 (2010)Google Scholar
  4. 4.
    Bonami P., Biegler L.T., Conn A.R., Cornuéjols G., Grossmann I.E., Laird C.D., Lee J., Lodi A., Margot F., Sawaya N., Wächter A.: An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optim. 5, 186–2004 (2008)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Callegari S., Bizzarri F., Rovatti R., Setti G.: On the approximate solution of a class of large discrete quadratic programming problems by ΔΣ modulation: The case of circulant quadratic forms. IEEE Trans. Signal Process. 58(12), 6126–6139 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dash, S., Lodi, A., Rajan, D.: {−1, 0, 1} unconstrained quadratic programs using max-flow based relaxations. Technical Report OR/05/13, DEIS, University of Bologna (2005)Google Scholar
  7. 7.
    De Simone C.: The cut polytope and the boolean quadric polytope. Discrete Math. 79, 71–75 (1989)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Eisenbrand F. : Integer programming and algorithmic geometry of numbers. In: Jünger, M., Liebling, T., Naddef, D., Nemhauser, G., Pulleyblank, W., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds) 50 Years of Integer Programming 1958–2008. The Early Years and State-of-the-Art Surveys, Springer, Berlin (2009)Google Scholar
  9. 9.
    Fincke U., Pohst M.: Improved methods for calculating vectors of short length in a lattice, including a complexity analysis. Math. Comput. 44, 463–471 (1985)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Frangioni A., Lodi A., Rinaldi G.: Optimizing over semimetric polytopes. In: Bienstock, D., Nemhauser, G. (eds) Integer Programming and Combinatorial Optimization—IPCO 2004, Volume 3064 of Lecture Notes in Computer Science, pp. 431–443. Springer, Berlin (2004)CrossRefGoogle Scholar
  11. 11.
    Hemmecke R., Köppe M., Lee J., Weismantel R.: Nonlinear integer programming. In: Jünger, M., Liebling, T., Naddef, D., Nemhauser, G., Pulleyblank, W., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds) 50 Years of Integer Programming 1958–2008. The Early Years and State-of-the-Art Surveys, Springer, Berlin (2009)Google Scholar
  12. 12.
    ILOG, Inc. ILOG CPLEX 12.1 (2009). http://www.ilog.com/products/cplex
  13. 13.
    Lodi A.: MIP computation and beyond. In: Jünger, M., Liebling, T., Naddef, D., Nemhauser, G., Pulleyblank, W., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds) 50 Years of Integer Programming 1958–2008. The Early Years and State-of-the-Art Surveys, Springer, Berlin (2009)Google Scholar
  14. 14.
    Lodi A., Allemand K., Liebling T.M.: An evolutionary heuristic for quadratic 0–1 programming. Eur. J. Oper. Res. 119, 662–670 (1999)MATHCrossRefGoogle Scholar
  15. 15.
    Micciancio D., Goldwasser S.: Complexity of Lattice Problems: A Cryptographic Perspective. Springer, Berlin (2002)MATHCrossRefGoogle Scholar
  16. 16.
    Moré J.J., Toraldo G.: On the solution of large quadratic programming problems with bound constraints. SIAM J. Optim. 1, 93–113 (1991)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Schnorr, C.P., Euchner, M.: Lattice basis reduction: improved practical algorithms and solving subset sum problems. In: 8th International Symposium on Fundamentals of Computation Theory, Volume 529 of Lecture Notes in Computer Science, pp. 68–85 (1991)Google Scholar
  18. 18.
    Shoup, V.: NTL: A Library for Doing Number Theory (version 5.5.2) (2009). http://www.shoup.net/ntl
  19. 19.
    Van Valkenburg M.E.: Analog Filter Design. The Oxford Series in Electrical and Computer Engineering. Oxford University Press, Oxford (1996)Google Scholar
  20. 20.
    Vallentin, F.: SHVEC: Shortest and Closest Vectors in Lattices (2006). http://www.fma2.math.uni-magdeburg.de/~latgeo/SHVEC-1.0/shvec-1.0.tar.gz

Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  • Christoph Buchheim
    • 1
  • Alberto Caprara
    • 2
  • Andrea Lodi
    • 2
  1. 1.Fakultät für MathematikTechnische Universität DortmundDortmundGermany
  2. 2.DEIS, University of BolognaBolognaItaly

Personalised recommendations