Table of Contents
ISRN Operations Research
Volume 2013 (2013), Article ID 393482, 7 pages
http://dx.doi.org/10.1155/2013/393482
Research Article

A Spline Smoothing Newton Method for Distance Regression with Bound Constraints

Li Dong1,2 and Bo Yu1

1School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China
2College of Science, Dalian Nationalities University, Dalian, Liaoning 116605, China

Received 15 February 2013; Accepted 19 March 2013

Academic Editors: I. Ahmad and X.-M. Yuan

Copyright © 2013 Li Dong and Bo Yu. 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. P. T. Boggs, R. H. Byrd, and R. B. Schnabel, “A stable and efficient algorithm for nonlinear orthogonal distance regression,” SIAM Journal on Scientific Computing, vol. 8, pp. 1052–1078, 1987. View at Google Scholar
  2. R. Strebel, D. Sourlier, and W. Gander, “A comparison of orthogonal least squares fitting in coordinate metrology,” in Recent Advances in Total Least Squares and Errors-in-Variables Techniques, S. Van Huffel, Ed., pp. 249–258, SIAM, Philadelphia, Pa, USA, 1997. View at Google Scholar
  3. J. W. Zwolak, P. T. Boggs, and L. T. Watson, “Algorithm 869: ODRPACK95: a weighted orthogonal distance regression code with bound constraints,” ACM Transactions on Mathematical Software, vol. 33, no. 4, Article ID 1268782, 2007. View at Publisher · View at Google Scholar · View at Scopus
  4. I. Al-Subaihi and G. A. Watson, “Fitting parametric curves and surfaces by l distance regression,” BIT Numerical Mathematics, vol. 45, no. 3, pp. 443–461, 2005. View at Publisher · View at Google Scholar · View at Scopus
  5. N. Z. Shor, Minimization Methods for Non-Differentiable Functions, Springer, New York, NY, USA, 1985.
  6. W. Murray and L. M. Overton, “A projected Lagrangian algorithm for nonlinear minimax optimization,” SIAM Journal on Scientific Computing, vol. 1, pp. 345–370, 1980. View at Google Scholar
  7. R. S. Womersley and R. Fletcher, “An algorithm for composite nonsmooth optimization problems,” Journal of Optimization Theory and Applications, vol. 48, no. 3, pp. 493–523, 1986. View at Publisher · View at Google Scholar · View at Scopus
  8. J. L. Zhou and A. L. Tits, “An SQP algorithm for finely discretized continuous minimax problems and other minimax problems with many objective functions,” SIAM Journal on Optimization, vol. 6, no. 2, pp. 461–487, 1996. View at Google Scholar · View at Scopus
  9. A. Frangioni, “Generalized bundle methods,” SIAM Journal on Optimization, vol. 13, no. 1, pp. 117–156, 2003. View at Publisher · View at Google Scholar · View at Scopus
  10. M. Gaudioso and M. F. Monaco, “A bundle type approach to the unconstrained minimization of convex nonsmooth functions,” Mathematical Programming, vol. 23, no. 2, pp. 216–226, 1982. View at Google Scholar · View at Scopus
  11. K. C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, vol. 1133 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1985.
  12. J. Zowe, “Nondifferentiable optimization: a motivation and a short introduction into the sub-gradient and the bundle concept,” in Computational Mathematical Programming, K. Schittkowski, Ed., vol. 15 of NATO SAI Series, pp. 323–356, Springer, New York, NY, USA, 1985. View at Google Scholar
  13. J. W. Bandler and C. Charalambous, “Practical least pth optimization of networks,” IEEE Transactions on Microwave Theory and Techniques, vol. 20, no. 12, pp. 834–840, 1972. View at Google Scholar · View at Scopus
  14. C. Charalambous, “Acceleration of the least pth algorithm for minimax optimization with engineering applications,” Mathematical Programming, vol. 17, no. 1, pp. 270–297, 1979. View at Publisher · View at Google Scholar · View at Scopus
  15. C. Gfgola and S. Gomez, “A regularization method for solving the finite convex min-max problem,” The SIAM Journal on Numerical Analysis, vol. 27, pp. 1621–1634, 1990. View at Google Scholar
  16. D. Q. Mayne and E. Polak, “Nondifferential optimization via adaptive smoothing,” Journal of Optimization Theory and Applications, vol. 43, no. 4, pp. 601–613, 1984. View at Publisher · View at Google Scholar · View at Scopus
  17. E. Polak, J. O. Royset, and R. S. Womersley, “Algorithms with adaptive smoothing for finite minimax problems,” Journal of Optimization Theory and Applications, vol. 119, no. 3, pp. 459–484, 2003. View at Publisher · View at Google Scholar · View at Scopus
  18. R. A. Polyak, “Smooth optimization methods for minimax problems,” SIAM Journal on Control and Optimization, vol. 26, no. 6, pp. 1274–1286, 1988. View at Google Scholar · View at Scopus
  19. S. Xu, “Smoothing method for minimax problems,” Computational Optimization and Applications, vol. 20, no. 3, pp. 267–279, 2001. View at Publisher · View at Google Scholar · View at Scopus
  20. I. Zang, “A smoothing technique for minCmax optimization,” Mathematics Programs, vol. 19, pp. 61–77, 1980. View at Google Scholar
  21. Y. Xiao and B. Yu, “A truncated aggregate smoothing Newton method for minimax problems,” Applied Mathematics and Computation, vol. 216, no. 6, pp. 1868–1879, 2010. View at Publisher · View at Google Scholar · View at Scopus
  22. G. H. Zhao, Z. R. Wang, and H. N. Mou, “Uniform approximation of min/max functions by smooth splines,” Journal of Computational and Applied Mathematics, vol. 236, pp. 699–703, 2011. View at Google Scholar
  23. L. Dong and B. Yu, “A spline smoothing Newton method for finite minimax problems,” preprint.
  24. E. Polak, Optimization: Algorithms and Consistent Approximations, Springer, New York, NY, USA, 1997.
  25. L. Dong, B. Yu, and G. H. Zhao, “A smoothing spline homotopy method for nonconvex nonlinear programming,” preprint.
  26. B. Yu, G. C. Feng, and S. L. Zhang, “The aggregate constraint homotopy method for nonconvex nonlinear programming,” Nonlinear Analysis: Theory, Methods and Applications, vol. 45, no. 7, pp. 839–847, 2001. View at Publisher · View at Google Scholar · View at Scopus
  27. Y. Xiao, B. Yu, and D. L. Wang, “Truncated smoothing newton method for l fitting rotated cones,” Journal of Mathematical Research and Exposition, vol. 30, pp. 159–166, 2010. View at Google Scholar
  28. H. Späth, “Least squares fitting with rotated paraboloids,” Mathematical Communications, vol. 6, no. 2, pp. 173–179, 2001. View at Google Scholar