Annals of Operations Research

, Volume 275, Issue 2, pp 587–605 | Cite as

Improved row-by-row method for binary quadratic optimization problems

  • Rupaj Kumar NayakEmail author
  • Nirmalya Kumar Mohanty
Original Research


The research presented here is an improved row-by-row (RBR) algorithm for the solution of boolean quadratic programming (BQP) problems. While a faster and implementable RBR method has been widely used for semidefinite programming (SDP) relaxed BQPs, it can be challenged by SDP relaxations because of the fact that it produce a tighter lower bounds than RBR on BQPs. On the other hand, solving SDP by interior point method (IPM) is computationally expensive for large scale problems. Departing from IPM, our methods provides better lower bound than the RBR algorithm by Wai et al. (IEEE international conference on acoustics, speech and signal processing ICASSP, 2011) and competitive with SDP solved by IPM. The method includes the SDP cut relaxation on the SDP and is solved by a modified RBR method. The algorithm has been tested on MATLAB platform and applied to several BQPs from BQPLIB (a library by the authors). Numerical experiments show that the proposed method outperform the previous RBR method proposed by several authors and the solution of BQP by IPM as well.


Unconstrained quadratic binary optimization Row-by-row method Semidefinite programming Computer vision First order method 


  1. Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  2. Burer, S., & Monteiro, R. D. C. (2003). A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Mathematical Programming, 95, 329–357.CrossRefGoogle Scholar
  3. Cour, T., & Bo, J. (2007). Solving Markov random fields with spectral relaxation. In Proceedings of the international conference on artificial intelligence and statistics.Google Scholar
  4. Goldfarb, D., Ma, S., & Wen, Z. (2009). Solving low-rank matrix completion problems efficiently. In IEEE conference on communication and control.Google Scholar
  5. Grant, M., & Boyd, S. (2014). CVX: Matlab software for disciplined convex programming, version 2.1.
  6. Grant, M., & Boyd, S. (2008). Graph implementations for nonsmooth convex programs. In V. Blondel, S. Boyd, & H. Kimura (Eds.), Recent advances in learning and control, Lecture notes in control and information sciences (pp. 95–110). New York: Springer.Google Scholar
  7. Grippo, L., & Sciandrone, M. (2000). On the convergence of the block nonlinear Gauss–Seidel method under convex constraints. Operations Research Letters, 26, 127–136.CrossRefGoogle Scholar
  8. Guattery, S., & Miller, G. (1998). On the quality of spectral separators. SIAM Journal on Matrix Analysis and Applications, 19, 701–719.CrossRefGoogle Scholar
  9. Heiler, M., Keuchel, J., & Schnörr, C. (2005). Semidefinite clustering for image segmentation with a-priori knowledge. In Pattern recognition (27th DAGM Symposium) (Vol. 3663 pp. 309–317). Springer.Google Scholar
  10. Joulin, A., Bach, F., & Ponce, J. (2010). Discriminative clustering for image cosegmentation. In Proceedings of the IEEE conference on computer vision.Google Scholar
  11. Kannan, R., Vempala, S., & Vetta, A. (2004). On clusterings: Good, bad and spectral. Journal of the ACM, 51, 497–515.CrossRefGoogle Scholar
  12. Keuchel, J., Schnörr, C., Schellewald, C., & Cremers, D. (2003). Binary partitioning, perceptual grouping and restoration with semidefinite programming. IEEE Transaction on Pattern Analysis and Machine Intelligence, 25, 1364–1379.CrossRefGoogle Scholar
  13. Kochenberger, G. A., Glover, F., Alidaee, B., & Rego, C. (2005). An unconstrained quadratic binary programming approach to the vertex coloring problem. Annals of Operations Research, 139(1), 229–241.CrossRefGoogle Scholar
  14. Lang, K. J. (2005). Fixing two weaknesses of the spectral method. In: Proceedings of advanced neural information processing systems (pp. 715–722).Google Scholar
  15. Lauer, F., & Schnörr, C. (2009). Spectral clustering of linear subspaces for motion segmentation. In Proceedings of the international conference on computer vision.Google Scholar
  16. Malick, J. (2007). The spherical constraint in boolean quadratic programs. Journal of Global Optimization, 39, 609–622.CrossRefGoogle Scholar
  17. Nayak, R. K., & Mohanty, N. K. (2017). Bqplib: A library for BQP.
  18. Nayak, R. K., & Biswal, M. P. (2018). A low complexity semidefinite relaxation for large-scale mimo detection. Journal of Combinatorial Optimization, 35(2), 473–492.CrossRefGoogle Scholar
  19. Olsson, C., Eriksson, A., & Kahl, F. (2007). Solving large scale binary quadratic problems: Spectral methods vs. semidefinite programming. In Proc. IEEE Conf. Comput. Vis. and Pattern Recogn (pp. 1–8).Google Scholar
  20. Pardalos, P. M. R. (1990). Parallel branch and bound algorithms for quadratic zero-one programs on the hypercube architecture. Annals of Operations Research, 22(1), 271–292.CrossRefGoogle Scholar
  21. Powell, M. J. D. (1973). On search directions for minimization algorithms. Mathematical Programming, 4, 193–201.CrossRefGoogle Scholar
  22. Schellewald, C., & Schnorr, C. (2005). Probabilistic subgraph matching based on convex relaxation. In Proceedings of the international conference on energy minimization methods in computer visible and pattern recoginition (pp. 171–186).Google Scholar
  23. Shi, J., & Malik, J. (2000). Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22, 888–905.CrossRefGoogle Scholar
  24. Srebro, N. (2004). Learning with matrix factorizations. Ph.D. thesis, Massachusetts Institute of Technology.Google Scholar
  25. Sturm, J. F. (1999). Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimizations Methods and Software, 11, 625–653.CrossRefGoogle Scholar
  26. Toh, K. C., Todd, M. J., & Tutuncu, R. H. (1999). Sdpt3—A Matlab software package for semidefinite programming (pp. 545–581).Google Scholar
  27. Tutuncu, R. H., Toh, K. C., & Todd, M. J. (2003). Solving semidefinite-quadratic-linear programs using SDPT3. Mathematical Programming Series B, 95, 189–217.CrossRefGoogle Scholar
  28. Wai, H. T., Ma, W. K., & So, M. C. A. (2011). Cheap semidefinite relaxation MIMO detection using row-by-row block coordinate descent. In IEEE international conference on acoustics, speech and signal processing ICASSP (pp. 3256–3259).Google Scholar
  29. Wang, P., Shen, C., & Hengel, A. V. D. (2013). A fast semidefinite approach to solving binary quadratic problems. In Proceedings of the IEEE conference on computer vision and pattern recognition.Google Scholar
  30. Wen, Z., Goldfarb, D., Ma, S., & Scheinberg, K. (2009). Row by Row methods for semidefinite programming. Technical report, Department of IEOR, Columbia University.Google Scholar
  31. Wen, Z., & Yin, W. (2013). A feasible method for optimization with orthogonality constraints. Mathematical Programming, 142, 397–434.CrossRefGoogle Scholar
  32. Yu, S. X., & Shi, J. (2004). Segmentation given partial grouping constraints. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26, 173–183.CrossRefGoogle Scholar
  33. Zhang, F. (2005). The Schur complement and its applications. New York: Springer.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.International Institute of Information TechnologyBhubaneswarIndia

Personalised recommendations