Advertisement

Feasible and Accurate Algorithms for Covering Semidefinite Programs

  • Garud Iyengar
  • David J. Phillips
  • Cliff Stein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6139)

Abstract

In this paper we describe an algorithm to approximately solve a class of semidefinite programs called covering semidefinite programs. This class includes many semidefinite programs that arise in the context of developing algorithms for important optimization problems such as Undirected Sparsest Cut, wireless multicasting, and pattern classification. We give algorithms for covering SDPs whose dependence on ε is ε − 1. These algorithms, therefore, have a better dependence on ε than other combinatorial approaches, with a tradeoff of a somewhat worse dependence on the other parameters. For many reasons, including numerical stability and a variety of implementation concerns, the dependence on ε is critical, and the algorithms in this paper may be preferable to those of the previous work. Our algorithms exploit the structural similarity between packing and covering semidefinite programs and packing and covering linear programs.

Keywords

Lagrangian Relaxation Semidefinite Program Saddle Point Problem Interior Point Algorithm Accurate Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arora, S., Hazan, E., Kale, S.: Fast algorithms for approximate semidefinite programming using the multiplicative weights update method. In: Proceedings of the 46th Annual Symposium on Foundations of Computer Science (2005)Google Scholar
  2. 2.
    Arora, S., Kale, S.: A combinatorial, primal-dual approach to semidefinite programs. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing (2007)Google Scholar
  3. 3.
    Arora, S., Rao, S., Vazirani, U.: Expander flows, geometric embeddings, and graph partitionings. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 222–231 (2004)Google Scholar
  4. 4.
    Bienstock, D.: Potential function methods for approximately solving linear programming problems: theory and practice, Boston, MA. International Series in Operations Research & Management Science, vol. 53 (2002)Google Scholar
  5. 5.
    Bienstock, D., Iyengar, G.: Solving fractional packing problems in \({O}^{*}(\frac{1}{\epsilon})\) iterations. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 146–155 (2004)Google Scholar
  6. 6.
    Fleischer, L.: Fast approximation algorithms for fractional covering problems with box constraint. In: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (2004)Google Scholar
  7. 7.
    Garg, N., Konemann, J.: Faster and simpler algorithms for multicommodity flow and other fractional packing problems. In: Proceedings of the 39th Annual Symposium on Foundations of Computer Science, pp. 300–309 (1998)Google Scholar
  8. 8.
    Goldberg, A.V., Oldham, J.D., Plotkin, S.A., Stein, C.: An implementation of a combinatorial approximation algorithm for minimum-cost multicommodity flow. In: Bixby, R.E., Boyd, E.A., Ríos-Mercado, R.Z. (eds.) IPCO 1998. LNCS, vol. 1412, pp. 338–352. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  9. 9.
    Iyengar, G., Phillips, D.J., Stein, C.: Approximation algorithms for semidefinite packing problems with applications to maxcut and graph coloring. In: Proceedings of the 11th Conference on Integer Programming and Combinatorial Optimization, pp. 152–166 (2005); Submitted to SIAM Journal on OptimizationGoogle Scholar
  10. 10.
    Iyengar, G., Phillips, D.J., Stein, C.: Feasible and accurate algorithms for covering semidefinite programs. Tech. rep., Optimization online (2010)Google Scholar
  11. 11.
    Klein, P., Lu, H.-I.: Efficient approximation algorithms for semidefinite programs arising from MAX CUT and COLORING. In: Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, PA, pp. 338–347. ACM, New York (1996)Google Scholar
  12. 12.
    Klein, P., Young, N.: On the number of iterations for Dantzig-Wolfe optimization and packing-covering approximation algorithms. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds.) IPCO 1999. LNCS, vol. 1610, pp. 320–327. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  13. 13.
    Lu, Z., Monteiro, R., Yuan, M.: Convex optimization methods for dimension reduction and coefficient estimation in multivariate linear regression, Arxiv preprint arXiv:0904.0691 (2009)Google Scholar
  14. 14.
    Nesterov, Y.: Smooth minimization of nonsmooth functions. Mathematical Programming 103, 127–152 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Nesterov, Y.: Smoothing technique and its applications in semidefinite optimization. Mathematical Programming 110, 245–259 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Nesterov, Y., Nemirovski, A.: Interior-point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics, vol. 13. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994)zbMATHGoogle Scholar
  17. 17.
    Plotkin, S., Shmoys, D.B., Tardos, E.: Fast approximation algorithms for fractional packing and covering problems. Mathematics of Operations Research 20, 257–301 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Sidiropoulos, N., Davidson, T., Luo, Z.: Transmit beamforming for physical-layer multicasting. IEEE Transactions on Signal Processing 54, 2239 (2006)CrossRefGoogle Scholar
  19. 19.
    Weinberger, K., Saul, L.: Distance metric learning for large margin nearest neighbor classification. The Journal of Machine Learning Research 10, 207–244 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Garud Iyengar
    • 1
  • David J. Phillips
    • 2
  • Cliff Stein
    • 1
  1. 1.The Department of Industrial Engineering & Operations ResearchColumbia University 
  2. 2.Mathematics DepartmentThe College of William & Mary 

Personalised recommendations