In this paper we present a generic primal-dual interior point methods (IPMs) for linear optimization in which the search direction depends on a univariate kernel function which is also used as proximity measure in the analysis of the algorithm. The proposed kernel function does not satisfy all the conditions proposed in [2]. We show that the corresponding large-update algorithm improves the iteration complexity with a factor ${n}^{\frac{1}{6}}$ when compared with the method based on the use of the classical logarithmic barrier function. For small-update interior point methods the iteration bound is $O(\sqrt{n}log\frac{n}{\u03f5}),$ which is currently the best-known bound for primal-dual IPMs.

Keywords: linear optimization, primal-dual interior-point algorithm, large and small-update method

@article{RO_2008__42_2_199_0, author = {Ghami, M. El and Roos, C.}, title = {Generic primal-dual interior point methods based on a new kernel function}, journal = {RAIRO - Operations Research - Recherche Op\'erationnelle}, pages = {199--213}, publisher = {EDP-Sciences}, volume = {42}, number = {2}, year = {2008}, doi = {10.1051/ro:2008009}, mrnumber = {2431399}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ro:2008009/} }

TY - JOUR AU - Ghami, M. El AU - Roos, C. TI - Generic primal-dual interior point methods based on a new kernel function JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2008 SP - 199 EP - 213 VL - 42 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ro:2008009/ DO - 10.1051/ro:2008009 LA - en ID - RO_2008__42_2_199_0 ER -

%0 Journal Article %A Ghami, M. El %A Roos, C. %T Generic primal-dual interior point methods based on a new kernel function %J RAIRO - Operations Research - Recherche Opérationnelle %D 2008 %P 199-213 %V 42 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ro:2008009/ %R 10.1051/ro:2008009 %G en %F RO_2008__42_2_199_0

Ghami, M. El; Roos, C. Generic primal-dual interior point methods based on a new kernel function. RAIRO - Operations Research - Recherche Opérationnelle, Volume 42 (2008) no. 2, pp. 199-213. doi : 10.1051/ro:2008009. http://archive.numdam.org/articles/10.1051/ro:2008009/

[1] Cs. Mészáros, and X. Xu, Implementation of interior point methods for large scale linear programming. In Interior Point Methods of Mathematical Programming, edited by T. Terlaky, Kluwer Academic Publishers, The Netherlands (1996) 189-252. | MR | Zbl

, ,[2] A comparative study of new barrier functions for primal- dual interior-point algorithms in linear optimization. SIAM J. Optim. 15 (2004) 101-128. | MR | Zbl

, , and ,[3] A new efficient large-update primal-dual interior-point method based on a finite barrier. SIAM J. Optim. 13 (2003) 766-782. | MR | Zbl

, , and ,[4] A primal-dual interior-point method for linear optimization based on a new proximity function. Optim. Methods Softw. 17 (2002) 985-1008. | MR | Zbl

, , and ,[5] Path following methods for linear programming. SIAM Rev. 34 (1992) 167-227. | MR | Zbl

,[6] Interior Point Approach to Linear, Quadratic and Convex Programming, Mathematics and its Applications 277. Kluwer Academic Publishers, Dordrecht, 1994. | MR | Zbl

,[7] Pathways to the optimal set in linear programming. In Progress in Mathematical Programming: Interior Point and Related Methods, edited by N. Megiddo, Springer Verlag, New York, 1989, 131-158. Identical version in: Proceedings of the 6th Mathematical Programming Symposium of Japan, Nagoya, Japan, (1986) 1-35. | MR | Zbl

,[8] Interior-path following primal-dual algorithms: Part I: Linear programming. Math. Program. 44 (1989) 27-41. | MR | Zbl

and ,[9] A new class of polynomial primal-dual methods for linear and semidefinite optimization. To appear in Eur. J. Oper. Res. | MR | Zbl

, , and ,[10] A new and efficient large-update interior-point method for linear optimization. J. Comput. Tech. 6 (2001) 61-80. | MR | Zbl

, , and ,[11] Self-regular functions and new search directions for linear and semidefinite optimization. Math. Program. 93 (2002) 129-171. | MR | Zbl

, , and ,[12] Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms. Princeton University Press, 2002. | MR | Zbl

, , and ,[13] Theory and Algorithms for Linear Optimization. An Interior-Point Approach. John Wiley & Sons, Chichester, UK, 1997. | MR | Zbl

, , and ,[14] An “analytic center” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. In System Modelling and Optimization : Proceedings of the 12th IFIP- Conference held in Budapest, Hungary, September 1985, edited by A. Prékopa, J. Szelezsán, and B. Strazicky, Lecture Notes in Control and Information Sciences, Springer Verlag, Berlin, West-Germany, 84 (1986) 866-876. | MR | Zbl

,[15] Recent developments and new directions in linear programming. In Mathematical Programming: Recent Developments and Applications, edited by M. Iri and K. Tanabe, Kluwer Academic Press, Dordrecht (1989) 109-157. | MR | Zbl

,[16] Interior Point Algorithms, Theory and Analysis, John Wiley and Sons, Chichester, UK, 1997. | MR | Zbl

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