A NEW PRIMAL-DUAL INTERIOR POINT METHOD FOR LINEAR OPTIMIZATION

  • Cho, Gyeong-Mi (DEPARTMENT OF MULTIMEDIA ENGINEERING, DONGSEO UNIVERSITY)
  • Received : 2009.02.10
  • Accepted : 2009.03.06
  • Published : 2009.03.25

Abstract

A primal-dual interior point method(IPM) not only is the most efficient method for a computational point of view but also has polynomial complexity. Most of polynomialtime interior point methods(IPMs) are based on the logarithmic barrier functions. Peng et al.([14, 15]) and Roos et al.([3]-[9]) proposed new variants of IPMs based on kernel functions which are called self-regular and eligible functions, respectively. In this paper we define a new kernel function and propose a new IPM based on this kernel function which has $O(n^{\frac{2}{3}}log\frac{n}{\epsilon})$ and $O(\sqrt{n}log\frac{n}{\epsilon})$ iteration bounds for large-update and small-update methods, respectively.

Acknowledgement

Supported by : Korea Research Foundation

References

  1. K. Amini and A. Haseli, A new proximity function generating the best known iteration bounds for both largeupdate and small-update interior point methods, ANZIAM J. 49 (2007), 259-270. https://doi.org/10.1017/S1446181100012827
  2. E.D. Andersen, J. Gondzio, Cs. Meszaros, and X. Xu, Implementation of interior point methods for large scale linear programming, in: T. Terlaky(Ed.), Interior point methods of mathematical programming, Kluwer Academic Publisher, The Netherlands, 189-252, 1996.
  3. Y.Q. Bai, M. El Ghami, and C. Roos, A new efficient large-update primal-dual interior-point method based on a finite barrier, Siam J. on Optimization 13 (2003), 766-782.
  4. Y.Q. Bai, M. El Ghami, and C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization, Siam J. on Optimization 15 (2004), 101-128. https://doi.org/10.1137/S1052623403423114
  5. Y.Q. Bai and C. Roos, A primal-dual interior point method based on a new kernel function with linear growth rate, in: Proceedings of the 9th Australian Optimization Day, Perth, Australia, 2002.
  6. Y.Q. Bai and C. Roos, A polynomial-time algorithm for linear optimization based on a new simple kernel function, Optimization Methods and Software 18 (2003), 631-646. https://doi.org/10.1080/10556780310001639735
  7. Y.Q. Bai, G. Lesaja, C. Roos, G.Q.Wang, and M. El Ghami, A class of large-update and small-update primaldual interior-point algorithms for linear optimization, J. Optim. Theory and Appl., DOI 10.1007/s10957-008-9389-z., 2008.
  8. M. El Ghami, I. Ivanov, J.B.M. Melissen, C. Roos, and T. Steihaug, A polynomial-time algorithm for linear optimization based on a new class of kernel functions, Journal of Computational and Applied Mathematics, DOI 10.1016/j.cam.2008.05.027., 2008.
  9. M. El Ghami and C. Roos, Generic primal-dual interior pont methods based on a new kernel function, RAIRO-Oper. Res. 42 (2008), 199-213. https://doi.org/10.1051/ro:2008009
  10. C.C. Gonzaga, Path following methods for linear programming, Siam Review 34 (1992), 167-227. https://doi.org/10.1137/1034048
  11. D. den Hertog, Interior point approach to linear, quadratic and convex programming, Mathematics and its Applications 277, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.
  12. N.K. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984), 373-395. https://doi.org/10.1007/BF02579150
  13. M. Kojima, S. Mizuno, and A. Yoshise, A primal-dual interior-point algorithm for linear programming, in: N. Megiddo(Ed.), Progress in mathematical programming: Interior point and related methods, Springer-Verlag, New York, 29-47, 1989.
  14. J. Peng, C. Roos, and T. Terlaky, Self-regular functions and new search directions for linear and semidefinite optimization, Mathematical Programming 93 (2002), 129-171. https://doi.org/10.1007/s101070200296
  15. J. Peng, C. Roos, and T. Terlaky, Self-Regularity, A new paradigm for primal-dual interior-point algorithms, Princeton University Press, 2002.
  16. C. Roos, T. Terlaky, and J. Ph. Vial, Theory and algorithms for linear optimization, An interior approach, John Wiley & Sons, Chichester, U.K., 1997.
  17. G. Sonnevend, An analytic center for polyhedrons and new classes of global algorithms for liner (smooth, convex) programming, in: A. Prekopa, J. Szleezsan, and B. Strazicky(Ed.), System modeling and optimization : Proceeding of the 12th IFIP-Conference, Budapest, Hungary, September 1985, Volume 84, Lecture Notes in Control and Information Sciences, Springer Verlag, Berlin, West-Germany, 866-876, 1986.
  18. N.J. Todd, Recent developments and new directions in linear programming, in: M. Iri and K. Tanabe(Ed.), Mathematical Programming : Recent developments and applications, Kluwer Academic Publishers, Dordrecht, 109-157, 1989.
  19. S.J. Wright, Primal-dual interior-point methods, SIAM, Philadelphia, USA, 1997.
  20. Y. Ye, Interior-point algorithms, John Wiley & Sons, Chichester, UK, 1997.