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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
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)
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)
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)
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)
Fleischer, L.: Fast approximation algorithms for fractional covering problems with box constraint. In: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (2004)
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)
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)
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 Optimization
Iyengar, G., Phillips, D.J., Stein, C.: Feasible and accurate algorithms for covering semidefinite programs. Tech. rep., Optimization online (2010)
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)
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)
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)
Nesterov, Y.: Smooth minimization of nonsmooth functions. Mathematical Programming 103, 127–152 (2005)
Nesterov, Y.: Smoothing technique and its applications in semidefinite optimization. Mathematical Programming 110, 245–259 (2007)
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)
Plotkin, S., Shmoys, D.B., Tardos, E.: Fast approximation algorithms for fractional packing and covering problems. Mathematics of Operations Research 20, 257–301 (1995)
Sidiropoulos, N., Davidson, T., Luo, Z.: Transmit beamforming for physical-layer multicasting. IEEE Transactions on Signal Processing 54, 2239 (2006)
Weinberger, K., Saul, L.: Distance metric learning for large margin nearest neighbor classification. The Journal of Machine Learning Research 10, 207–244 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Iyengar, G., Phillips, D.J., Stein, C. (2010). Feasible and Accurate Algorithms for Covering Semidefinite Programs. In: Kaplan, H. (eds) Algorithm Theory - SWAT 2010. SWAT 2010. Lecture Notes in Computer Science, vol 6139. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13731-0_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-13731-0_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13730-3
Online ISBN: 978-3-642-13731-0
eBook Packages: Computer ScienceComputer Science (R0)