Skip to main content

On the bridge between combinatorial optimization and nonlinear optimization: a family of semidefinite bounds for 0–1 quadratic problems leading to quasi-Newton methods

Abstract

This article presents a family of semidefinite programming bounds, obtained by Lagrangian duality, for 0–1 quadratic optimization problems with linear or quadratic constraints. These bounds have useful computational properties: they have a good ratio of tightness to computing time, they can be optimized by a quasi-Newton method, and their final tightness level is controlled by a real parameter. These properties are illustrated on three standard combinatorial optimization problems: unconstrained 0–1 quadratic optimization, heaviest \(k\)-subgraph, and graph bisection.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

References

  1. 1.

    Anjos, M., Lasserre, J.B.: Handbook of Semidefinite, Conic and Polynomial Optimization. In: International Series in Operations Research & Management Science, Vol. 166. Springer (2012)

  2. 2.

    Billionnet, A., Elloumi, S.: Using a mixed integer quadratic programming solver for the unconstrained quadratic 0–1 problem. Math. Program. 109(1), 55–68 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Billionnet, A., Elloumi, S., Plateau, M.-C.: Improving the performance of standard solvers for quadratic 0–1 programs by a tight convex reformulation: the QCR method. Discret. Appl. Math. 157(6), 1185–1197 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Bonnans, J.F., Gilbert, J.Ch., Lemaréchal, C., Sagastizábal, C.: Numerical Optimization. Springer, Berlin (2003)

  5. 5.

    Billionnet, A.: Different formulations for solving the heaviest k-subgraph problem. Inf. Syst. Oper. Res. 43(3), 171–186 (2005)

    MathSciNet  Google Scholar 

  6. 6.

    Borchers, B.: CSDP, a C library for semidefinite programming. Optim. Methods Softw. 11(1), 613–623 (1999)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

  8. 8.

    de Klerk, E., Sotirov, R.: Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem. Math. Prog. 122(2) (2010)

  9. 9.

    Faye, A., Roupin, F.: Partial lagrangian for general quadratic programming. 4’OR A Q. J. Oper. Res. 5(1), 75–88 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Gilbert, J.Ch., Lemaréchal, C.: Some numerical experiments with variable-storage quasi-Newton algorithms. Math. Program. 45, 407–435 (1989)

    Google Scholar 

  11. 11.

    Goemans, M., Williamson, D.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 6, 1115–1145 (1995)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Helmberg, C.: Semidefinite Programming for Combinatorial Optimization. PhD thesis, Habilitationsschrift, TU Berlin (2000)

  13. 13.

    Helmberg. C.: A C++ Implementation of the Spectral Bundle Method, Version 1.1.3. http://www-user.tu-chemnitz.de/~helmberg/SBmethod/ (2004)

  14. 14.

    Higham, N.: Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Appl 103, 103–118 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Helmberg, C., Poljak, S., Rendl, F., Wolkowicz, H.: Combining semidefinite and polyhedral relaxations for integer programs. In: Balas, E., Clausen, J. (eds.) IPCO, Volume 920 of Lecture Notes in Computer Science, pp. 124–134. Springer (1995)

  16. 16.

    Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM J. Optim. 10(3), 673–696 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Helmberg, C., Rendl, F., Vanderbei, R.J., Wolkowicz, H.: An interior point method for semidefinite programming. SIAM J. Optim. 6, 342–361 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Two volumes. Springer, Heidelberg (1993)

  19. 19.

    Kim, S., Kojima, M., Waki, H.: Exploiting sparsity in sdp relaxation for sensor network localization. SIAM J. Optim. 20(1), 192–215 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Karisch, S., Rendl, F., Clausen, J.: Solving graph bisection problems with semidefinite programming. INFORMS J Comput. 12(3), 177–191 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Lemaréchal, C.: Lagrangian relaxation. In: Junger, M., Naddef, D. (eds.) Computational Combinatorial Optimization, pp. 112–156. Springer, Heidelberg (2001)

  22. 22.

    Lemaréchal, C., Oustry, F.: Semidefinite relaxations and Lagrangian duality with application to combinatorial optimization. Rapport de Recherche 3710, INRIA (1999)

  23. 23.

    Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Malick, J.: Spherical constraint in Boolean quadratic programming. J. Glob. Optim. 39(4) (2007)

  25. 25.

    Malick, J., Povh, J., Rendl, F., Wiegele, A.: Regularization methods for semidefinite programming. SIAM J. Optim. 20(1), 336–356 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Malick, J., Roupin, F.: Numerical study of semidefinite bounds for the k-cluster problem. In: n ISCO’10, International Symposium on Combinatorial Optimization. Electronics Notes of Discrete Mathematics, pp. 399–406. Elsevier (2010)

  27. 27.

    Malick, J., Roupin, F.: Solving k-cluster problems to optimality with semidefinite programming. Math. Program. 136(2), 279–300 (2012)

    Google Scholar 

  28. 28.

    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)

    MATH  Book  Google Scholar 

  29. 29.

    Pardalos, P.., Iasemidis, L.D, Sackellares, J.C., Chaovalitwongse, W., Carney, P., Prokopyev, O., Yatsenko, V., Shiau, D.-S.: Seizure warning algorithm based on optimization and nonlinear dynamics. Math. Program. 101:365–385 (2004)

    Google Scholar 

  30. 30.

    Pardalos, P., Rodgers, G.P.: Computational aspects of a branch and bound algorithm for quadratic zero-one programming. Computing 45, 134–144 (1990)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Poljak, S., Rendl, F., Wolkowicz, H.: A recipe for semidefinite relaxation for (0,1)-quadratic programming. J. Glob. Optim. 7, 51–73 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Qi, L.Q., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58(3), 353–367 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Roupin, F.: From linear to semidefinite programming: an algorithm to obtain semidefinite relaxations for bivalent quadratic problems. J. Comb. Optim. 8(4), 469–493 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyedral relaxations. Math. Program. 121, 307–335 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discret. Math. 3(3), 411–430 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Saigal, R., Vandenberghe, L., Wolkowicz, H.: Handbook of Semidefinite Programming. Kluwer, Dordrecht (2000)

Download references

Acknowledgments

We are grateful to Sourour Elloumi who provided us a lot of material for the numerical tests, and to Sofia Zaourar who helped us to develop parts of the solver for bisection problems.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jérôme Malick.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Malick, J., Roupin, F. On the bridge between combinatorial optimization and nonlinear optimization: a family of semidefinite bounds for 0–1 quadratic problems leading to quasi-Newton methods. Math. Program. 140, 99–124 (2013). https://doi.org/10.1007/s10107-012-0628-6

Download citation

Keywords

  • Lagrangian duality
  • Combinatorial optimization
  • 0–1 quadratic programming
  • Nonlinear programming
  • Semidefinite programming
  • Quasi-Newton

Mathematics Subject Classification

  • 90C22
  • 90C27
  • 90C57