Continuous Relaxation for Discrete DC Programming

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 359)


Discrete DC programming with convex extensible functions is studied. A natural approach for this problem is a continuous relaxation that extends the problem to a continuous domain and applies the algorithm in continuous DC programming. By employing a special form of continuous relaxation, which is named “lin-vex extension,” the optimal solution of the continuous relaxation coincides with the original discrete problem. The proposed method is demonstrated for the degree-concentrated spanning tree problem.


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  1. 1.
    Fujishige, S.: Submodular Functions and Optimization, 2nd edn. Annals of Discrete Mathematics, vol. 58. Elsevier, Amsterdam (2005)MATHGoogle Scholar
  2. 2.
    Horst, R., Thoai, N.V.: DC Programming: Overview. Journal of Optimization Theory and Applications 103, 1–43 (1999)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Iyer, R., Jegelka, S., Bilmes, J.: Fast semidifferential-based submodular function optimization. In: Proceedings of the 30th International Conference on Machine Learning, pp. 855–863 (2013)Google Scholar
  4. 4.
    Kawahara, Y., Washio, T.: Prismatic algorithm for discrete D.C. programming problem. In: Proceedings of the 25th Annual Conference on Neural Information Processing Systems, pp. 2106–2114 (2011)Google Scholar
  5. 5.
    Kobayashi, Y.: The complexity of maximizing the difference of two matroid rank functions. METR2014-42, University of Tokyo (2014)Google Scholar
  6. 6.
    Korte, B., Vygen, J.: Combinatorial Optimization, 5th edn. Springer, Berlin (2012)CrossRefGoogle Scholar
  7. 7.
    Lemke, P.: The maximum leaf spanning tree problem for cubic graphs is NP-complete. IMA Preprint Series #428, University of Minnesota (1988)Google Scholar
  8. 8.
    Le-Thi, H.A.: DC Programming and DCA, (retrieved at February 23, 2015)
  9. 9.
    Maehara, T., Murota, K.: A framework of discrete DC programming by discrete convex analysis. Mathematical Programming (2014),
  10. 10.
    Moriguchi, S., Shioura, A., Tsuchimura, N.: M-convex function minimization by continuous relaxation approach—Proximity theorem and algorithm. SIAM Journal on Optimization 21, 633–668 (2011)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Moriguchi, S., Tsuchimura, N.: Discrete L-convex function minimization based on continuous relaxation. Pacific Journal of Optimization 5, 227–236 (2009)MATHMathSciNetGoogle Scholar
  12. 12.
    Murota, K.: Discrete Convex Analysis. Society for Industrial and Applied Mathematics, Philadelphia (2003)CrossRefMATHGoogle Scholar
  13. 13.
    Murota, K.: Recent developments in discrete convex analysis. In: Cook, W., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, ch. 11, pp. 219–260. Springer, Berlin (2009)Google Scholar
  14. 14.
    Narasimhan, M., Bilmes, J.: A submodular-supermodular procedure with applications to discriminative structure learning. In: Proceedings of the 21st Conference on Uncertainty in Artificial Intelligence, pp. 404–412 (2005)Google Scholar
  15. 15.
    Tanenbaum, A.S.: Computer Networks, 5th edn. Prentice Hall, Upper Saddle River (2010)Google Scholar
  16. 16.
    Pham Dinh, T., Le Thi, H.A.: Convex analysis approach to D.C. programming: Theory, algorithms and applications. Acta Mathematica Vietnamica 22, 289–355 (1997)MATHMathSciNetGoogle Scholar
  17. 17.
    Hoang, T.: D.C. optimization: Theory, methods and algorithms. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 149–216. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar
  18. 18.
    Yuille, A.L., Rangarajan, A.: The concave-convex procedure. Neural Computation 15, 915–936 (2003)CrossRefMATHGoogle Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematical and Systems EngineeringShizuoka UniversityShizuokaJapan
  2. 2.Department of Mathematical InformaticsUniversity of TokyoTokyoJapan

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