Skip to main content
SpringerLink
Log in

Computational experience with a bundle approach for semidefinite cutting plane relaxations of Max-Cut and Equipartition

  • Published:
Mathematical Programming Submit manuscript

Abstract

We propose a dynamic version of the bundle method to get approximate solutions to semidefinite programs with a nearly arbitrary number of linear inequalities. Our approach is based on Lagrangian duality, where the inequalities are dualized, and only a basic set of constraints is maintained explicitly. This leads to function evaluations requiring to solve a relatively simple semidefinite program. Our approach provides accurate solutions to semidefinite relaxations of the Max-Cut and the Equipartition problem, which are not achievable by direct approaches based only on interior-point methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alizadeh, F.: Interior Point methods in semidefinite programming with applications to combinatorial optimization. SIAM Journal on Optimization 5, 13–51 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  2. Atkinson, D.S., Vaidya, P.M.: A cutting plane algorithm for convex programming that uses analytic centers. Mathematical Programming 69, 1–43 (1995)

    MATH  MathSciNet  Google Scholar 

  3. Bahiense, L., Maculan, N., Sagastizabal, C.: The volume algorithm revisited: relation with bundle methods. Mathematical Programming 94, 41–69 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Balas, C., Christofides, N.: A restricted Lagrangian approach to the traveling salesman problem. Mathematical Programming 21, 19–46 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barahona, F., Anbil, R.: The Volume algorithm: producing primal solutions with a subgradient method. Mathematical Programming 87, 385–399 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barahona, F., Jünger, M., Reinelt, G.: Experiments in quadratic 0-1 programming. Mathematical Programming 44, 127–137 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  7. Barahona, F., Ladanyi, L.: Branch and Cut based on the Volume Algorithm: Steiner trees in graphs and Max-Cut. Technical report, (2001)

  8. Belloni, A., Sagastizabal, C.: Dynamic bundle methods: applications to combinatorial optimization problems. Technical report, (2004)

  9. Billionnet, A., Elloumi, S.: Using a mixed integer quadratic programming solver for the unconstrained quadratic 0-1 problem. Technical report, CNAM, rapport technique 466, (2003)

  10. Burer, S., Monteiro, R.D.C., Zhang, Y.: Rank-two relaxation heuristics for Max-Cut and other binary quadratic programs. SIAM Journal on Optimization 12, 503–521 (2002)

    MathSciNet  Google Scholar 

  11. Burer, S., Monteiro, R.D.C., Zhang, Y.: A computational study of gradient-based log-barrier algorithms for a class of large-scale sdps. Mathematical Programming 95, 359–379 (2003)

    MATH  MathSciNet  Google Scholar 

  12. Choi, C., Ye, Y.: Solving sparse semidefinite programs using the dual scaling algorithm with an iterative solver. Technical report, University of Iowa, (2000)

  13. Delorme, C., Poljak, S.: Laplacian eigenvalues and the maximum cut problem. Mathematical Programming 62, 557–574 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  14. Donath, W.E., Hoffman, A.J.: Lower bounds for the partitioning of graphs. IBM J. of Research and Developement 17, 420–425 (1973)

    MATH  MathSciNet  Google Scholar 

  15. Frangioni, A.: Generalized bundle methods. SIAM Journal on Optimization 13, 117–156 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42, 1115–1145 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  17. Goffin, J.L., Haurie, A., Vial, J.P.: Decomposition and nondifferentiable optimization with the projective algorithm. Management Science 38, 284–302 (1992)

    MATH  Google Scholar 

  18. Held, M., Karp, R.: The traveling salesman problem and minimum spanning trees. Operations Research 18, 1138–1162 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  19. Helmberg, C.: An interior point method for semidefinite programming and max-cut bounds. PhD thesis, Graz University of Technology, Austria, (1994)

  20. Helmberg, C., Kiwiel, K.C., Rendl, F.: Incorporating inequality constraints in the spectral bundle method. In: Boyd, E.A., Bixby, R.E., Rios-Mercado, R.Z., (ed.) Integer Programming and combinatorial optimization, 423–435. Springer Lecture Notes 1412, (1998)

  21. Helmberg, C., Rendl, F.: Solving quadratic (0,1)-problems by semidefinite programming and cutting planes. Mathematical Programming 82, 291–315 (1998)

    MATH  MathSciNet  Google Scholar 

  22. Helmberg, C., Rendl, F., Vanderbei, R., Wolkowicz, H.: An interior-point method for semidefinite programming. SIAM Journal on Optimization 6, 342–361 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  23. Karisch, S.E., Rendl, F.: Semidefinite Programming and Graph Equipartition. Fields Institute Communications 18, 77–95 (1998)

    MATH  MathSciNet  Google Scholar 

  24. Kiwiel, K.C.: Methods of descent for nondifferentiable optimization. Springer, Berlin, (1985)

  25. Lemarechal, C.: Nonsmooth optimization and descent methods. Technical report, International Institute for Applied Systems Analysis, (1978)

  26. Lemarechal, C.: The omnipresence of Lagrange. 4OR 1, 7–25 (2003)

  27. Lemarechal, C., Nemirovskii, A., Nesterov, Y.: New variants of bundle methods. Mathematical Programming 69, 111–147 (1995)

    MATH  MathSciNet  Google Scholar 

  28. Lisser, A., Rendl, F.: Graph partitioning using Linear and Semidefinite Programming. Mathematical Programming (B) 95, 91–101 (2002)

    MathSciNet  Google Scholar 

  29. Lovász, L.: On the shannon capacity of a graph. IEEE Trans. Inform. Theory 25, 1–7 (1979)

    MATH  MathSciNet  Google Scholar 

  30. Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM Journal on Optimization 1, 166–190 (1991)

    MATH  MathSciNet  Google Scholar 

  31. Meurdesoif, P.: Strengthening the Lovasz bound for graph coloring. Mathematical Programming (forthcoming).

  32. Nesterov, Y.: Complexity estimates of some cutting plane methods based on the analytic barrier. Mathematical Programming 69, 149–176 (1995)

    MATH  MathSciNet  Google Scholar 

  33. Nesterov, Y.: Quality of semidefinite relaxation for nonconvex quadratic optimization. Technical report, (1997)

  34. Pardalos, P.M., Rodgers, G.P.: Computational aspects of a branch and bound algorithm for quadratic zero-one programming. Computing 45, 131–144 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  35. Poljak, S., Rendl, F.: Nonpolyhedral relaxations of graph bisection problems. SIAM Journal on Optimization 5, 467–487 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  36. Polyak, B.T.: A general method for solving extremum problems. Soviet Mathematics 8, 593–597 (1966)

    Google Scholar 

  37. Rendl, F., Sotirov, R.: Bounds for the quadratic assignment problem using the bundle method. Technical report, University of Klagenfurt, Austria, (2003)

  38. Schramm, H., Zowe, J.: A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results. SIAM J. Optimization 2, 121–152 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  39. Sotirov, R.: The bundle method in combinatorial optimization. PhD thesis, University of Klagenfurt, Austria, (2003)

  40. Todd, M.J.: A study of search directions in primal-dual interior-point methods for semidefinite programming. Optimization Methods and Software 11, 1–46 (1999)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität Wien, Fakultät für Mathematik, Nordbergstraße 15, 1090, Wien, Austria

    Ilse Fischer

  2. Department of Geoinformation, Carinthia Tech Institute, Europastrasse 4, 9524, Villach, Austria

    Gerald Gruber

  3. Institut für Mathematik, Universität Klagenfurt, Universitätsstraße 65-67, 9020, Klagenfurt, Austria

    Franz Rendl

  4. Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    Renata Sotirov

Authors
  1. Ilse Fischer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gerald Gruber
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Franz Rendl
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Renata Sotirov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Franz Rendl.

Additional information

Received: April, 2004

The last author gratefully acknowledges the support from the Austrian Science Foundation FWF Project P12660-MAT.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fischer, I., Gruber, G., Rendl, F. et al. Computational experience with a bundle approach for semidefinite cutting plane relaxations of Max-Cut and Equipartition. Math. Program. 105, 451–469 (2006). https://doi.org/10.1007/s10107-005-0661-9

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-005-0661-9

Keywords

Mathematics Subject Classification (2000)

Search

Navigation