Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-20T04:15:13.875Z Has data issue: false hasContentIssue false
  • English
  • Français

An interior point method for linear programming based on a class of Kernel functions

Published online by Cambridge University Press:  17 April 2009

K. Amini  and
M. R. Peyghami
K. Amini
Affiliation:
Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran, e-mail: amini@mehr.sharif.edu, peyghami@mehr.sharif.edu
M. R. Peyghami
Affiliation:
Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran, e-mail: amini@mehr.sharif.edu, peyghami@mehr.sharif.edu
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Interior point methods are not only the most effective methods for solving optimisation problems in practice but they also have polynomial time complexity. However, there is still a gap between the practical behavior of the interior point method algorithms and their theoretical complexity results. In this paper, by focusing on linear programming problems, we introduce a new family of kernel functions that have some simple and easy to check properties. We present a simplified analysis to obtain the complexity of generic interior point methods based on the proximity functions induced by these kernel functions. Finally, we prove that this family of kernel functions leads to improved iteration bounds of the large-update interior point methods.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 1 , February 2005 , pp. 139 - 153
Copyright
Copyright © Australian Mathematical Society 2005

References

[1] Andersen, E.D., Gondzio, J., Mészáros, Cs. and Xu, X., ‘Implementation of interior point methods for large scale linear programming’, in Interior Point Methods of Mathematical Programming, (Terlaky, T., Editor) (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996), pp. 189–252.CrossRefGoogle Scholar
[2]Bai, Y.Q., Ghami, M. El and Roos, C., ‘A new efficient large-update primal-dual interior-point method based on a finite barrier’, SIAM J. Optim. (electronic) 13 (2003), 766–782.Google Scholar
[3]Karmarkar, N.K., ‘A new polynomial time algorithm for linear programming’, Combinatorica 4 (1984), 373–395.CrossRefGoogle Scholar
[4] Megiddo, N., ‘Pathways to the optimal set in linear programming’, in Progress in Mathematical Programming: Interior Point and Related Methods, (Megiddo, N., Editor) (Springer-Verlag, New York, 1989), pp. 131–158.Google Scholar
[5]Peng, J., Roos, C. and Terlaky, T., ‘A new class of polynomial primal-dual methods for linear and semidefinite optimization’, European J. Oper. Res. 143 (2002), 234–256.CrossRefGoogle Scholar
[6]Peng, J., Roos, C. and Terlaky, T., ‘A new and efficient large-update interior-point method for linear optimization’, Vychisl. Technol. 6 (2001), 61–80.Google Scholar
[7]Peng, J., Roos, C. and Terlaky, T., ‘Self-regular functions and new search directions for linear and semidefinite optimization’, Math. Program. 93 (2002), 129–171.CrossRefGoogle Scholar
[8]Peng, J., Roos, C. and Terlaky, T., Self-regularity: A new paradigm for primal-dual interior-point algorithms, Princeton Series in Applied Mathematics (Princeton University Press, Princeton, N.J., 2002).Google Scholar
[9]Roos, C., Terlaky, T. and Vial, J.-Ph., Theory and algorithms for linear optimization: An interior-point approach (John Wiley & Sons, Chichester, UK, 1997).Google Scholar
[10] Sonnevend, G., ‘An “analytic center” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming’, (Prékopa, A., Szelezsán, J., and Strazicky, B., Editors), Lecture Notes in Control and Information Sciences 84 (Springer-Verlag, Berlin, 1986), pp. 866–876.Google Scholar
[11]Wright, S.J., Primal-dual interior-point methods (SIAM, Philadelphia, USA, 1997).Google Scholar
[12]Ye, Y., Interior point algorithms, theory and analysis (John Wiley & Sons, Chichester, UK, 1997).CrossRefGoogle Scholar