Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-28T03:56:19.034Z Has data issue: false hasContentIssue false
  • English
  • Français

A new proximity function generating the best known iteration bounds for both large-update and small-update interior-point methods

Published online by Cambridge University Press:  17 February 2009

Keyvan Aminis  and
Arash Haseli
Keyvan Aminis
Affiliation:
department of Sciences Razi University, Kermanshah Iran; email: keyvanamini1353@yahoo.com or kamini@razi.ac.ir.
Arash Haseli
Affiliation:
Islamic Azad University Kermanshah branch, Kermanshah Iran; email: arhaseli@yahoo.com.
Rights & Permissions [Opens in a new window]

Abstract

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 (IPMs) are not only very effective in practice for solving linear optimization problems but also have polynomial-time complexity. Despite the practical efficiency of large-update algorithms, from a theoretical point of view, these algorithms have a weaker iteration bound with respect to small-update algorithms. In fact, there is a significant gap between theory and practice for large-update algorithms. By introducing self-regular barrier functions, Peng, Roos and Terlaky improved this gap up to a factor of log n. However, checking these self-regular functions is not simple and proofs of theorems involving these functions are very complicated. Roos el al. by presenting a new class of barrier functions which are not necessarily self-regular, achieved very good results through some much simpler theorems. In this paper we introduce a new kernel function in this class which yields the best known complexity bound, both for large-update and small-update methods.

Keywords

linear optimizationinterior-point methodprimal-dual methodkernel functionlarge-updatesmall-updatepolynomial complexity.
Type
Articles
Information
The ANZIAM Journal , Volume 49 , Issue 2 , October 2007 , pp. 259 - 270
Copyright
Copyright © Australian Mathematical Society 2007

References

[1] Bai, Y. Q., Elghami, M. and Roos, C., “A new efficient large-update primal-dual interior-point method based on a finite barrier”, SIAM J Optim. 13 (2003) 766–782 (electronic).CrossRefGoogle Scholar
[2] Bai, Y. Q., Elghami, M. and Roos, C., “A comparative study of new barrier functions for primal-dual interior-point algorithms in linear optimization”, SIAM J Optim. 15 (2004) 101–128.CrossRefGoogle Scholar
[3] Bai, Y. Q., Guo, J. and Roos, C., A new kernel function yielding the best known iteration bounds for primal-dual interior-point algorithms, 2006, working paper, available at http://www.isa.ewi.tudelft.nl/roos/wpapers.html.Google Scholar
[4] Megiddo, N., Pathways to the optimal set in linear programming, in Progress in Mathematical Programming: Interior Point and Related Methods (ed. N., Megiddo), (Springer Verlag, New York, 1989), 131–158. (Identical version in: Proceedings of the 6th Mathematical Programming Symposium of Japan, Nagoya, Japan, pages 1–35, 1986.)CrossRefGoogle Scholar
[5] 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.Google Scholar
[6] Peng, J., Roos, C. and Terlaky, T., Self-Regularity. A New Paradigm for Primal-Dual Interior-Point Algorithms (Princeton University Press, Princeton, New Jersey, 2002).Google Scholar
[7] Roos, C.. Terlaky, T. and Vial, J. Ph., Theory and Algorithms for Linear Optimization. An Interior Point Approach (John Wiley & Sons, Chichester, 1997).Google Scholar