Cutting Planes for Mixed 0-1 Semidefinite Programs

  • G. Iyengar
  • M. T. Çezik
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2081)


Since the seminal work of Nemirovski and Nesterov [14], research on semidefinite programming (SDP) and its applications in optimization has been burgeoning. SDP has led to good relaxations for the quadratic assignment problem, graph partition, non-convex quadratic optimization problems, and the TSP. SDP-based relaxations have led to approximation algorithms for combinatorial optimization problems such as the MAXCUT and vertex coloring. SDP has also found nu- merous applications in robust control and, as a natural extension, in robust op- timization for convex programs with uncertain parameters. For a recent survey of semidefinite techniques and applications see [20].


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. Alizadeh. Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim., 5(1):13–51, 1995.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    E. Balas. Disjunctive programming. Annals of Discrete Mathematics, 5:3–51, 1979.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    E. Balas, S. Ceria, and G. Cornuèjols. A list-and-project cutting plane algorithm for mixed 0-1 programs. Mathematical Programming, 58:295–324, 1993.CrossRefMathSciNetGoogle Scholar
  4. 4.
    E. Balas, S. Ceria, and G. Cornuèjols. Mixed 0-1 programming by lift-and-project in a branch-and-cut framework. Management Science, 42:1229–1246, 1996.MATHCrossRefGoogle Scholar
  5. 5.
    S. Benson, Y. Ye, and X. Zhang. Solving large-scale sparse semidefinite programs for combinatorial optimization. Tech. Rep., Dept of Mgmt Sc., Univ. of Iowa, 1997.Google Scholar
  6. 6.
    M. T. Ç ezik. Semidefinite methods for the traveling salesman and other combinatorial problems. PhD thesis, IEOR Dept., Columbia University, 2000.Google Scholar
  7. 7.
    C. Choi and Y. Ye. Solving sparse semidefinite programs using dual scaling algorithm with an iterative solver. Tech. Rep., Dept. of Mgmt. Sc., Univ. of Iowa, 2000.Google Scholar
  8. 8.
    D. Cvetković, M. Čangalović, and V. Kovačević-Vujčić. Semidenite programming methods for the symmetric TSP. In IPCO VII, p. 126–136. Springer, Berlin, 1999.Google Scholar
  9. 9.
    M. Goemans and F. Rendl. Combinatorial optimization. In Handbook of Semidefinite Programming: Theory, Algorithms and Applications. Kluwer, 2000.Google Scholar
  10. 10.
    M. Goemans and D. P. Williamson. Improved approximation algorithms for maximum cut and satis ability problems using semidefinite programming. Journal of ACM, 42:1115–1145, 1995.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    L. Lovász. Combinatorial Problems and Exercises. North-Holland, 1979.Google Scholar
  12. 12.
    L. Lovász and A. Schrijver. Cones of matrices and set-functions and 0-1 optimization. SIAM Journal of Control and Optimization, 1(2):166–190, 1991.MATHCrossRefGoogle Scholar
  13. 13.
    J. E. Mitchell and B. Borchers. Solving linear ordering problems with a combined interior point/simplex cutting plane algorithm. In High performance optimization, pages 349–366. Kluwer Acad. Publ., Dordrecht, 2000.Google Scholar
  14. 14.
    Y. Nesterov and A. Nemirovskii. Interior-point polynomial algorithms in convex programming. SIAM, Philadelphia, 1993.Google Scholar
  15. 15.
    G. Pataki. Cone-lp’s and semidefinite programs: geometry and simplex-type method. In Lecture Notes in Computer Science, v. 1084, p. 162–174, 1996.Google Scholar
  16. 16.
    H. Sherali and W. Adams. A heirarchy of relaxations and convex hull representations for mixed zero-one programming problems. Tech. Rep., Virginia Tech., 1989.Google Scholar
  17. 17.
    R. A. Stubbs and S. Mehrotra. A branch-and-cut method for 0-1 mixed convex programming. Mathematical Programming, 86:515–532, 1999.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    J. Sturm. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software, 11-12:625–653, 1999.CrossRefMathSciNetGoogle Scholar
  19. 19.
    TSPLIB: A library of sample instances for the TSP and related problems. Available at
  20. 20.
    W. Wolkowicz, R. Saigal, and L. Vandenberghe, editors. Handbook of Semidefinite Programming. Kluwer Acad. Publ., 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • G. Iyengar
    • 1
  • M. T. Çezik
    • 2
  1. 1.IEOR DeptColumbia UniversityNYUSA
  2. 2.Bell LaboratoriesLucent TechnologiesHolmdelUSA

Personalised recommendations