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Journal of Applied Mathematics
Volume 2012, Article ID 946893, 21 pages
http://dx.doi.org/10.1155/2012/946893
Research Article

An Interior Point Method for Solving Semidefinite Programs Using Cutting Planes and Weighted Analytic Centers

1Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN 55455, USA
2Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, Arizona 86011-5717, USA

Received 11 October 2011; Revised 10 May 2012; Accepted 24 May 2012

Academic Editor: James Buchanan

Copyright © 2012 John Machacek and Shafiu Jibrin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. Jibrin, Redundancy in Semidefinite Programming: Detection and Elimination of Redundant Linear Matrix Inequalities, VDM, Saarbrucken, Germany, 2009.
  2. L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Review, vol. 38, no. 1, pp. 49–95, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. L. Vandenberghe and S. Boyd, “Applications of semidefinite programming,” Applied Numerical Mathematics, vol. 29, no. 3, 1999. View at Google Scholar
  4. F. Alizadeh, “Interior point methods in semidefinite programming with applications to combinatorial optimization,” SIAM Journal on Optimization, vol. 5, no. 1, pp. 13–51, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, New York, NY, USA, 2004.
  6. R. J. Caron, T. Traynor, and S. Jibrin, “Feasibility and constraint analysis of sets of linear matrix inequalities,” INFORMS Journal on Computing, vol. 22, no. 1, pp. 144–153, 2010. View at Publisher · View at Google Scholar
  7. J. W. Chinneck, “The constraint consensus method for finding approximately feasible points in nonlinear programs,” INFORMS Journal on Computing, vol. 16, no. 3, pp. 255–265, 2004. View at Publisher · View at Google Scholar
  8. W. Ibrahim and J. W. Chinneck, “Improving solver success in reaching feasibility for sets of nonlinear constraints,” Computers & Operations Research, vol. 35, no. 5, pp. 1394–1411, 2008. View at Publisher · View at Google Scholar
  9. R. E. Gomory, “Outline of an algorithm for integer solutions to linear programs,” Bulletin of the American Mathematical Society, vol. 64, no. 5, pp. 275–278, 1958. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. J. E. Kelley, Jr., “The cutting-plane method for solving convex programs,” Journal of the Society for Industrial and Applied Mathematics, vol. 8, no. 4, pp. 703–712, 1960. View at Google Scholar
  11. E. W. Cheney and A. A. Goldstein, “Newton's method for convex programming and Tchebycheff approximation,” Numerische Mathematik, vol. 1, no. 1, pp. 253–268, 1959. View at Publisher · View at Google Scholar
  12. J. Renegar, “A polynomial-time algorithm, based on Newton's method, for linear programming,” Mathematical Programming, vol. 40, no. 1–3, pp. 59–93, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. I. S. Pressman and S. Jibrin, “The weighted analytic center for linear matrix inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol. 2, no. 3, article 29, 2002. View at Google Scholar · View at Zentralblatt MATH
  14. C. B. Chua, “A new notion of weighted centers for semidefinite programming,” SIAM Journal on Optimization, vol. 16, no. 4, pp. 1092–1109, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. S. Jibrin and J. W. Swift, “The boundary of weighted analytic centers for linear matrix inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol. 5, no. 1, article 14, 2004. View at Google Scholar · View at Zentralblatt MATH
  16. J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, NY, USA, 2nd edition, 2006.
  17. K. C. Tutuncu and M. J. Todd, On the Implementation and Usage of SDPT3-a Matlab Software Package for Semidefinite-Quadratic-Linear Programming Version 4, 2006.
  18. B. Borchers and L. Vandenberghe, “SDPLIB 1.2, a library of semidefinite programming test problems,” Optimization Methods and Software, vol. 11-12, no. 1–4, pp. 683–690, 1999. View at Publisher · View at Google Scholar