Fixing variables in semidefinite relaxations

  • Christoph Heimberg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1284)


The standard technique of reduced cost fixing from linear programming is not trivially extensible to semidefinite relaxations as the corresponding Lagrange multipliers are usually not available. We pro pose a general technique for computing reasonable Lagrange multipliers to constraints which are not part of the problem description. Its specialization to the semidefinite {-1,1} relaxation of quadratic 0-1 programming yields an efficient routine for fixing variables. The routine offers the possibility to exploit problem structure. We extend the traditional bijective map between {0,1} and {-1,1} formulations to the constraints such that the dual variables remain the same and structural properties are preserved. In consequence the fixing routine can efficiently be applied to optimal solutions of the semidefinite {0,1} relaxation of constrained quadratic 0-1 programming, as well. We provide numerical results showing the efficacy of the approach.

Key words

semidefinite programming semidefinite relaxations quadratic 0-1 programming reduced cost fixing AMS subject classifications. 90C31, 90C25, 65K10, 49K40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. Alizadeh, J.-P. A. Haeberly, and M. L. Overton. Coplementarity and nondegeneracy in semidefinite programming. Mathematical Programming, 77(2):111–128, 1997.Google Scholar
  2. 2.
    F. Barahona, M. Jünger, and G. Reinelt. Experiments in quadratic 0-1 programming. Mathematical Programming, 44:127–137, 1989.Google Scholar
  3. 3.
    C. De Simone. The cut polytope and the boolean quadric polytope. Discrete Applied Mathematics, 79:71–75, 1989.Google Scholar
  4. 4.
    M. X. Goemans and D. P. Williamson. Improved approxiamtion algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42:1115–1145, 1995.Google Scholar
  5. 5.
    G. H. Golub and C. F. van Loan. Matrix Computations. The Johns Hopkins University Press, 2nd edition, 1989.Google Scholar
  6. 6.
    M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, 2nd edition, 1988.Google Scholar
  7. 7.
    C. Helmberg, S. Poljak, F. Rendl, and H. Wolkowicz. Combining semidefinite and polyhedral relaxations for integer programs. In E. Balas and J. Clausen, editors, Integer Programming and Combinatorial Optimization, volume 920 of Lecture Notes in Computer Science, pages 124–134. Springer, May 1995.Google Scholar
  8. 8.
    C. Helmberg and F. Rendl. Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. ZIB Preprint SC-95-35, Konrad Zuse Zentrum für Informationstechnik Berlin, Takustraße 7, D-14195 Dahlem, Germany, Nov. 1995.Google Scholar
  9. 9.
    S. E. Karisch and F. Rendl. Semidefinite programming and graph equipartition. Technical Report 302, Department of Mathematics, Graz University of Technology, Graz, Austria, Dec. 1995.Google Scholar
  10. 10.
    M. Laurent, S. Poljak, and F. Rendl. Connections between semidefinite relaxations of the max-cut and stable set problems. Mathematical Programming, 77(2):225–246, 1997.Google Scholar
  11. 11.
    L. Lovász. On the Shannon capacity of a graph. IEEE Transactions on Information Theory, IT-25(1):1–7, Jan. 1979.Google Scholar
  12. 12.
    L. Lovász and A. Schrijver. Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optimization, 1(2):166–190, May 1991.Google Scholar
  13. 13.
    Y. Nesterov and M. J. Todd. Self-scaled barriers and interior-point methods for convex programming. Technical Report TR 1091, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York 14853, Apr. 1994. Revised June 1995, to appear in Mathematics of Operations Research.Google Scholar
  14. 14.
    L. Tunçel. Primal-dual symmetry and scale invariance of interior-point algorithms for convex optimization. CORR Report 96-18, Department of Combinatorics and Optimization, Univeristy of Waterloo, Ontario, Canada, Nov. 1996.Google Scholar
  15. 15.
    A.C. Williams. Quadratic 0-1 programming using the roof dual with computational results. RUTCOR Research Report 8-85, Rutgers Unversity, 1985.Google Scholar
  16. 16.
    H. Wolkowicz and Q. Zhao. Semidefinite programming relaxations for the graph partitioning problem. Corr report, University of Waterloo, Ontario, Canada, Oct. 1996.Google Scholar
  17. 17.
    Q. Zhao, S. E. Karisch, F. Rendl, and H. Wolkowicz. Semidefinite programming relaxations for the quadratic assignment problem. CORR Report 95/27, University of Waterloo, Ontario, Canada, Sept. 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Christoph Heimberg
    • 1
  1. 1.Konrad-Zuse-Zentrum für Informationstechnik BerlinBerlinGermany

Personalised recommendations