Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2013, Special Issue (2013), Article ID 963563, 7 pages.

A Rank-Two Feasible Direction Algorithm for the Binary Quadratic Programming

Xuewen Mu and Yaling Zhang

Full-text: Open access

Abstract

Based on the semidefinite programming relaxation of the binary quadratic programming, a rank-two feasible direction algorithm is presented. The proposed algorithm restricts the rank of matrix variable to be two in the semidefinite programming relaxation and yields a quadratic objective function with simple quadratic constraints. A feasible direction algorithm is used to solve the nonlinear programming. The convergent analysis and time complexity of the method is given. Coupled with randomized algorithm, a suboptimal solution is obtained for the binary quadratic programming. At last, we report some numerical examples to compare our algorithm with randomized algorithm based on the interior point method and the feasible direction algorithm on max-cut problem. Simulation results have shown that our method is faster than the other two methods.

Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2013), Article ID 963563, 7 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807674

Digital Object Identifier
doi:10.1155/2013/963563

Mathematical Reviews number (MathSciNet)
MR3127467

Zentralblatt MATH identifier
06950963

Citation

Mu, Xuewen; Zhang, Yaling. A Rank-Two Feasible Direction Algorithm for the Binary Quadratic Programming. J. Appl. Math. 2013, Special Issue (2013), Article ID 963563, 7 pages. doi:10.1155/2013/963563. https://projecteuclid.org/euclid.jam/1394807674


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References

  • C. Helmberg, Semidefinite Programming for Combinatorial Op-timization, Konrad-Zuse-Zentrum fur Information-Stechnik, Berlin, Germany, 2000.
  • F. Barahona, M. Groetschel, M. Juenger, and G. Reinelt, “An application of combinatorial optimization to statiscal optimization and circuit layout design,” Operations Research, vol. 36, no. 3, pp. 493–513, 1988.
  • P. H. Tan and L. K. Rasmussen, “The application of semidefinite programming for detection in CDMA,” IEEE Journal on Selected Areas in Communications, vol. 19, no. 8, pp. 1442–1449, 2001.
  • F. Hasegawa, J. Luo, K. Pattipati, and P. Willett, “Speed and accuracy comparation of techniques to solve a binary programming problem with application to syschronous CDMA,” IEEE Transaction on Communication, vol. 52, pp. 2775–2780, 2004.
  • X. Mu and Y. Zhang, “A new rank-two semidefinite programming relaxation method for multiuser detection problem,” Wireless Personal Communications, vol. 65, pp. 223–233, 2012.
  • J. Keuchel, C. Schnörr, C. Schellewald, and D. Cremers, “Binary partitioning, perceptual grouping, and restoration with semidefinite programming,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 25, no. 11, pp. 1364–1379, 2003.
  • S.-P. Wu, S. Boyd, and L. Vandenberghe, “FIR filter design via semidefinite programming and spectral factorization,” in Proceedings of the 35th IEEE Conference on Decision and Control, pp. 271–276, Kobe, Japan, December 1996.
  • M. X. Goemans and D. P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” Journal of the Association for Computing Machinery, vol. 42, no. 6, pp. 1115–1145, 1995.
  • S. Homer and M. Peinado, “Design and performance of parallel and distributed approximation algorithms for maxcut,” Journal of Parallel and Distributed Computing, vol. 46, no. 1, pp. 48–61, 1997.
  • S. Burer and R. D. C. Monteiro, “A projected gradient algorithm for solving the maxcut SDP relaxation,” Optimization Methods and Software, vol. 15, no. 3-4, pp. 175–200, 2001.
  • S. Burer, R. D. C. Monteiro, and Y. Zhang, “Rank-two relaxation heuristics for max-cut and other binary quadratic programs,” SIAM Journal on Optimization, vol. 12, no. 2, pp. 503–521, 2001.
  • S. Burer, R. D. C. Monteiro, and Y. Zhang, “Maximum stable set formulations and heuristics based on continuous optimization,” Mathematical Programming A, vol. 94, no. 1, pp. 137–166, 2002.
  • H. Liu, X. Wang, and S. Liu, “Feasible direction algorithm for solving the SDP relaxations of quadratic $\{-1,1\}$ programming problems,” Optimization Methods & Software, vol. 19, no. 2, pp. 125–136, 2004.
  • F. Alizadeh, P. A. Haeberly, and M. L. Overton, “Primal-dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results,” SIAM Journal on Optimization, vol. 8, no. 3, pp. 746–768, 1998.
  • M. J. Todd, “Semidefinite optimization,” Acta Numerica, vol. 10, pp. 515–560, 2001.
  • C.-x. Xu, X.-l. He, and F.-m. Xu, “An effective continuous algorithm for approximate solutions of large scale max-cut problems,” Journal of Computational Mathematics, vol. 24, no. 6, pp. 749–760, 2006.
  • F. Alizadeh, J. P. Haeberly, M. V. Nayakkankuppam, M. L. Overton, and S. Schmieta, “SDPpack user's guide-version 0.9Beta,” Tech. Rep. TR1997-737, Courant Institute of Mathematical Science, New York, NY, USA, June 1997.
  • J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11-12, no. 1–4, pp. 625–653, 1999.
  • C. Choi and Y. Ye, “Solving Sparse Semidefinite Programs Using the Dual Scaling Algorithm with an Iterative Solver,” Working Paper, Department of Management Science, University of Iowa, Iowa City, Iowa, USA, 2000.
  • G. Rinaldi, “Rudy graph generator,” http://www-user.tu- chemnitz.de/helmberg/rudy.tar.gz.
  • C. Helmberg and F. Rendl, “A spectral bundle method for semidefinite programming,” SIAM Journal on Optimization, vol. 10, no. 3, pp. 673–696, 2000.
  • H. Alperin and I. Nowak, “Lagrangian smoothing heuristics for max-cut,” Journal of Heuristics, vol. 11, no. 5-6, pp. 447–463, 2005.