Skip to main content
SpringerLink
Log in

An augmented Lagrangian method for binary quadratic programming based on a class of continuous functions

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper, an augmented Lagrangian method is proposed for binary quadratic programming (BQP) problems based on a class of continuous functions. The binary constraints are converted into a class of continuous functions. The approach reformulates the BQP problem as an equivalent augmented Lagrangian function, and then seeks its minimizer via an augmented Lagrangian method, in which the subproblem is solved by Barzilai–Borwein type method. Numerical results are reported for max-cut problem. The results indicate that the augmented Lagrangian approach is promising for large scale binary quadratic programming by the quality of the near optimal values and the low computational time.

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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Helmberg, C.: Semidefinite Programming for Combinatorial Optimization. Konrad-Zuse-Zentrum fur informationstechnik, Berlin (2000)

    Google Scholar 

  2. Goeman, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiably problem using semidefinite programming. J. ACM 42, 1115–1145 (1995)

    Article  Google Scholar 

  3. Barahon, F., Grotschel, M.: An application of combinatorial optimization to statiscal optimization and circuit layout design. Oper. Res. 36, 493–513 (1998)

    Article  Google Scholar 

  4. Peng, H.T., Rasmussen, L.K.: The application of semidefinite programming for detection in CDMA. IEEE Select. Commun. 19, 1442–1449 (2001)

    Article  Google Scholar 

  5. Hasegawa, F., Luo, J., Pattipati, K., Willett, P.: Speed and accuracy comparation of techniques to solve a binary programming problem with application to syschronous CDMA. IEEE Trans. Commun. 52, 2775–2780 (2004)

    Article  Google Scholar 

  6. Keuchel, J., Schnorr, C., Schellewald, C., Cremers, D.: Binary partitioning, perceptual grouping, and restoration with semidefinite programming. IEEE Trans. Pattern Anal. Mach. Intell. 25, 1364–1379 (2003)

    Article  Google Scholar 

  7. Olsson, C., Eriksson, A.P., Kahl, F.: Improved spectral relaxation methods for binary quadratic optimization problems. Comput. Vis. Image Understand. 112, 3–13 (2008)

    Article  Google Scholar 

  8. Wu, S.P., Boyd, S.: FIR filter design via semidefinite programming and spectral factorization. In: Proc. 35th Conf. Decision and Control, Kobe, pp. 271–276 (1996)

  9. Peinadoo, M., Homer, S.: Design and performance of parallel and distributed approximation algorithms for max-cut. J. Parallel Distrib. Comput. 9, 141–160 (1998)

    Google Scholar 

  10. Burer, S., Monteriro, R.D.C.: A projected gradient algorithm for solving the max-cut relaxation. Optim. Methods Softw. 15, 175–200 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Burer, S., Monteriro, R.D.C., Zhang, Y.: Rank-two relaxation heuristics for MAX-CUT and other binary quadratic programs. SIAM J. Optim. 12, 503–521 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu, H., Wang, X., Liu, S.: Feasible direction algorithm for solving SDP relaxation of the quadratic \(\{-1,1\}\) programming. Optim. Methods Softw. 19, 125–136 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Murray, W., Ng, K.M.: An algorithm for nonlinear optimization problems with binary variables. Comput. Optim. Appl. 47, 257–288 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pardalos, P.M.: Continuous approaches to discrete optimization problems. In: Di, G., Giannesi, F. (eds.) Nonlinear Optimization and Applications, pp. 313–328. Plenum, New York (1996)

    Chapter  Google Scholar 

  15. Polijak, S., Wolkowicz, H.: Convex relaxation of (0,1)-quadratic programming. Math. Oper. Res. 3, 550–561 (1995)

    Article  Google Scholar 

  16. Pan, S.H., Tan, T., Jiang, Y.: A global continuation algorithm for solving binary quadratic programming problems. Comput. Optim. Appl. 41, 349–362 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen, J.S., Li, J.F., Wu, J.: A continuation approach for solving binary quadratic program based on a class of NCP-functions. Appl. Math. Comput. 219, 3975–3992 (2012)

    Article  MathSciNet  Google Scholar 

  18. Liu, W.L.: Studies on algorithms for some problems in the field of communication signal processing. Doctor Dissertation, Department of Mathematics of Dalian University of Technology (2004)

  19. Todd, M.J.: Semidefinite optimization. Acta Numer. 10, 515–560 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Alizadeh, F., Haeberly, J.P., Overton, M.L.: Primal-dual interior point methods for semidefinite programming: convergence rates, stability, and numerical results. SIAM J. Optim. 8, 746–768 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Barzilai, J., Borwein, J.M.: Two point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)

    MATH  Google Scholar 

  23. Rinaldi, G.: Rudy graph generator (1998). http://www-user.tu-chemnitz.de/~helmberg/rudy.tar.gz. Accessed 12 Jan 2014

  24. Helmberg, C., Rendle, F.: A spectral bundle method for semidefinite programming. SIAM J. Optim. 10, 673–695 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Alperin, H., Nowak, I.: Lagrangian smoothing heuristics for max-cut. J. Heurist. 11, 447–463 (2005)

    Article  MATH  Google Scholar 

  26. Choi, C., Ye, Y.: Solving sparse semidefinite programs using the dual scaling algorithm with an iterative solver. Working paper, Department of Management Science, University of Iowa, IA (2000)

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, China

    Xuewen Mu

  2. School of Information and Communication Engineering, Dalian University of Technology, Dalian, 116024, China

    Wenlong Liu

Authors
  1. Xuewen Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenlong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuewen Mu.

Additional information

The work is supported by National Science Foundation for Young Scientists of China (No. 11101320), by the National Natural Science Foundation of China (No. 11032008, 11372069), and by China Scholarship Council (CSC). This work was conducted while the first author has been visiting Ohio University, Department of Mathematics.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mu, X., Liu, W. An augmented Lagrangian method for binary quadratic programming based on a class of continuous functions. Optim Lett 10, 485–497 (2016). https://doi.org/10.1007/s11590-015-0873-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-015-0873-8

Keywords

Search

Navigation