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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 598563, 10 pages
http://dx.doi.org/10.1155/2014/598563
Research Article

Proximal Alternating Direction Method with Relaxed Proximal Parameters for the Least Squares Covariance Adjustment Problem

1School of Mathematics and Physics, Changzhou University, Jiangsu 213164, China
2College of Science, Nanjing University of Posts and Telecommunications, Jiangsu 210003, China

Received 13 June 2013; Accepted 27 July 2013; Published 21 January 2014

Academic Editor: Abdellah Bnouhachem

Copyright © 2014 Minghua Xu et al. 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. R. Borsdorf and N. J. Higham, “A preconditioned Newton algorithm for the nearest correlation matrix,” IMA Journal of Numerical Analysis, vol. 30, no. 1, pp. 94–107, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. S. Boyd and L. Xiao, “Least-squares covariance matrix adjustment,” SIAM Journal on Matrix Analysis and Applications, vol. 27, no. 2, pp. 532–546, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. Y. Gao and D. Sun, “Calibrating least squares semidefinite programming with equality and inequality constraints,” SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 3, pp. 1432–1457, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  4. S. Gravel and V. Elser, “Divide and concur: a general approach to constraint satisfaction,” Physical Review E, vol. 78, Article ID 036706, 2008.
  5. N. J. Higham, “Computing a nearest symmetric positive semidefinite matrix,” Linear Algebra and its Applications, vol. 103, pp. 103–118, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. N. J. Higham, “Computing the nearest correlation matrix—a problem from finance,” IMA Journal of Numerical Analysis, vol. 22, no. 3, pp. 329–343, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. N. J. Higham, “Matrix nearness problems and applications,” in Applications of Matrix Theory, M. Gover and S. Barnett, Eds., vol. 22, pp. 1–27, Oxford University Press, Oxford, UK, 1989. View at Zentralblatt MATH · View at MathSciNet
  8. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, UK, 2004. View at MathSciNet
  9. 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 · View at MathSciNet
  10. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. H. Tütüncü, K. C. Toh, and M. J. Todd, “Solving semidefinite-quadratic-linear programs using SDPT3,” Mathematical Programming, vol. 95, no. 2, pp. 189–217, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Malick, “A dual approach to semidefinite least-squares problems,” SIAM Journal on Matrix Analysis and Applications, vol. 26, no. 1, pp. 272–284, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. H. Qi and D. Sun, “A quadratically convergent Newton method for computing the nearest correlation matrix,” SIAM Journal on Matrix Analysis and Applications, vol. 28, no. 2, pp. 360–385, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. B. He, M. Xu, and X. Yuan, “Solving large-scale least squares semidefinite programming by alternating direction methods,” SIAM Journal on Matrix Analysis and Applications, vol. 32, no. 1, pp. 136–152, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. P. M. Pardalos and M. G. C. Resende, Handbook of Applied Optimization, Oxford University Press, Oxford, UK, 2002. View at MathSciNet
  16. P. M. Pardalos, T. M. Rassias, and A. A. Khan, Nonlinear Analysis and Variational Problems, vol. 35 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2010, In honor of George Isac, Edited by Panos M. Pardalos, Themistocles M. Rassias and Akhtar A. Khan. View at Publisher · View at Google Scholar · View at MathSciNet
  17. H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol. 38, no. 3, pp. 367–426, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. W. Zhang, D. Han, and Z. Li, “A self-adaptive projection method for solving the multiple-sets split feasibility problem,” Inverse Problems, vol. 25, no. 11, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. W. Zhang, D. Han, and X. Yuan, “An efficient simultaneous method for the constrained multiple-sets split feasibility problem,” Computational Optimization and Applications, vol. 52, no. 3, pp. 825–843, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York, NY, USA, 1984. View at MathSciNet
  21. R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, vol. 9 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  22. J. Eckstein, “Some saddle-function splitting methods for convex programming,” Optimization Methods and Software, vol. 4, pp. 75–83, 1994.
  23. J. Eckstein and M. Fukushima, “Some reformulations and applications of the alternating direction method of multipliers,” in Large Scale Optimization: State of the Art, W. W. Hager, D. W. Hearn, and P. M. Pardalos, Eds., pp. 115–134, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994. View at Zentralblatt MATH · View at MathSciNet
  24. M. Fukushima, “Application of the alternating direction method of multipliers to separable convex programming problems,” Computational Optimization and Applications, vol. 1, no. 1, pp. 93–111, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. B. He and H. Yang, “Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities,” Operations Research Letters, vol. 23, no. 3-5, pp. 151–161, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. B. He, L.-Z. Liao, D. Han, and H. Yang, “A new inexact alternating directions method for monotone variational inequalities,” Mathematical Programming, vol. 92, no. 1, pp. 103–118, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. M. H. Xu, “Proximal alternating directions method for structured variational inequalities,” Journal of Optimization Theory and Applications, vol. 134, no. 1, pp. 107–117, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. D. Gabay, “Applications of the method of multipliers to variational inequalities,” in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, M. Fortin and R. Glowinski, Eds., pp. 299–331, North-Holland, Amsterdam, The Netherlands, 1983.
  29. D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximations,” Computer and Mathematics with Applications, vol. 2, pp. 17–40, 1976.
  30. O. Güler, “New proximal point algorithms for convex minimization,” SIAM Journal on Optimization, vol. 2, no. 4, pp. 649–664, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. W. W. Hager and H. Zhang, “Asymptotic convergence analysis of a new class of proximal point methods,” SIAM Journal on Control and Optimization, vol. 46, no. 5, pp. 1683–1704, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. W. W. Hager and H. Zhang, “Self-adaptive inexact proximal point methods,” Computational Optimization and Applications, vol. 39, no. 2, pp. 161–181, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM Journal on Control and Optimization, vol. 14, no. 5, pp. 877–898, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. M. Teboulle, “Convergence of proximal-like algorithms,” SIAM Journal on Optimization, vol. 7, no. 4, pp. 1069–1083, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. W. K. Glunt, “An alternating projections method for certain linear problems in a Hilbert space,” IMA Journal of Numerical Analysis, vol. 15, no. 2, pp. 291–305, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, NY, USA, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  37. N. Narendra, “A new polynomial time algorithm for linear programming,” Combinatorica, vol. 4, pp. 373–395, 1987.