Narendra K. Karmarkar is in the Mathematical Foundations of Computing Department and Jeffrey C. Lagarias is in the Mathematical Studies Department; they are with AT&T Bell Laboratories in Murray Hill, New Jersey.
Power Series Variants of Karmarkar-Type Algorithms
Article first published online: 29 JUL 2013
© 1989 AT&T Technical Journal
AT&T Technical Journal
Volume 68, Issue 3, pages 20–36, May-June 1989
How to Cite
Karmarkar, N. K., Lagarias, J. C., Slutsman, L. and Wang, P. (1989), Power Series Variants of Karmarkar-Type Algorithms. AT&T Technical Journal, 68: 20–36. doi: 10.1002/j.1538-7305.1989.tb00316.x
- Issue published online: 29 JUL 2013
- Article first published online: 29 JUL 2013
- Manuscript received February 14, 1989
Many interior-point linear programming algorithms have been proposed since the Karmarkar algorithm for linear programming problems appeared in 1984. These algorithms follow tangent (first-order) approximations to families of continuous trajectories that underlie such algorithms. This paper describes power-series variants of such algorithms that follow higher-order, truncated, power-series approximations to such trajectories. The choice of the power-series parameter is important to the performance of such algorithms, and this paper describes an apparently good choice of parameter. We describe two power-series algorithms; one builds on the dual-affine scaling algorithm and the other on a primal-dual path-following algorithm. Empirical results indicate that, compared to first-order methods, these higher-order power-series algorithms accelerate convergence by reducing the number of iterations. Both of these power-series algorithms have been successfully implemented in the AT&T KORBX® system.