# American Institute of Mathematical Sciences

January  2016, 12(1): 103-116. doi: 10.3934/jimo.2016.12.103

## A full-modified-Newton step infeasible interior-point algorithm for linear optimization

 1 Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018 2 Department of Mathematics, Zhejiang A&F University, Zhejiang, 311300, China, China

Received  August 2014 Revised  November 2014 Published  April 2015

Based on an equivalent reformulation of the central path, we obtain a modified-Newton step for linear optimization. Using this step, we propose an infeasible interior-point algorithm. The algorithm uses only one full-modified-Newton step search in each iteration. The complexity bound of the algorithm is the best known for infeasible interior-point algorithm.
Citation: Yinghong Xu, Lipu Zhang, Jing Zhang. A full-modified-Newton step infeasible interior-point algorithm for linear optimization. Journal of Industrial and Management Optimization, 2016, 12 (1) : 103-116. doi: 10.3934/jimo.2016.12.103
##### References:
 [1] G. Gu, H. Mansouri, M. Zangiabadi, Y. Bai and C. Roos, Improved full-Newton step $\mathcal O(nL)$ infeasible interior-point method for linear optimization, Journal of Optimization Theory and Applications, 145 (2010), 271-288. doi: 10.1007/s10957-009-9634-0. [2] H. Mansouri and C. Roos, Simplified $\mathcal O(nL)$ infeasible interior-point algorithm for linear optimization using full-newton steps, Optimization Methods and Software, 22 (2007), 519-530. doi: 10.1080/10556780600816692. [3] H. Mansouri, M. Zangiabadi, Y. Bai and C. Roos, An infeasible interior-point algorithm with full-Newton step for linear optimization, 2008. Available from: http://www.optimization-online.org/DB-HTML/2008/07/2052.html. [4] C. Roos, A full-Newton step $\mathcal O(n)$ infeasible interior-point algorithm for linear optimization, SIAM Journal on Optimization, 16 (2006), 1110-1136. doi: 10.1137/050623917. [5] C. Roos, An improved and simplified full-newton step $\mathcal O(n)$ infeasible interior-point method for linear optimization, SIAM J. Optim., 25 (2015), 102-114, Available from: http://www.optimization-online.org/DB-HTML/2014/01/4205.html. doi: 10.1137/140975462. [6] C. Roos, T. Terlaky and J. P. Vial, Interior Point Methods for Linear Optimization, Revised edition, Springer, New York, 2006. [7] S. Wright, Primal-dual Interior-point Methods, SIAM, Philadelphia, 1997. doi: 10.1137/1.9781611971453. [8] L. Zhang and Y. Xu, A new infeasible interior-point algorithm with full step for linear optimization based on a simple function, International Journal of Computer Mathematics, 88 (2011), 3163-3185. doi: 10.1080/00207160.2011.597503. [9] L. Zhang and Y. Xu, A full-newton step interior-point algorithm based on modified newton direction, Operations Research Letters, 39 (2011), 318-322. doi: 10.1016/j.orl.2011.06.006.

show all references

##### References:
 [1] G. Gu, H. Mansouri, M. Zangiabadi, Y. Bai and C. Roos, Improved full-Newton step $\mathcal O(nL)$ infeasible interior-point method for linear optimization, Journal of Optimization Theory and Applications, 145 (2010), 271-288. doi: 10.1007/s10957-009-9634-0. [2] H. Mansouri and C. Roos, Simplified $\mathcal O(nL)$ infeasible interior-point algorithm for linear optimization using full-newton steps, Optimization Methods and Software, 22 (2007), 519-530. doi: 10.1080/10556780600816692. [3] H. Mansouri, M. Zangiabadi, Y. Bai and C. Roos, An infeasible interior-point algorithm with full-Newton step for linear optimization, 2008. Available from: http://www.optimization-online.org/DB-HTML/2008/07/2052.html. [4] C. Roos, A full-Newton step $\mathcal O(n)$ infeasible interior-point algorithm for linear optimization, SIAM Journal on Optimization, 16 (2006), 1110-1136. doi: 10.1137/050623917. [5] C. Roos, An improved and simplified full-newton step $\mathcal O(n)$ infeasible interior-point method for linear optimization, SIAM J. Optim., 25 (2015), 102-114, Available from: http://www.optimization-online.org/DB-HTML/2014/01/4205.html. doi: 10.1137/140975462. [6] C. Roos, T. Terlaky and J. P. Vial, Interior Point Methods for Linear Optimization, Revised edition, Springer, New York, 2006. [7] S. Wright, Primal-dual Interior-point Methods, SIAM, Philadelphia, 1997. doi: 10.1137/1.9781611971453. [8] L. Zhang and Y. Xu, A new infeasible interior-point algorithm with full step for linear optimization based on a simple function, International Journal of Computer Mathematics, 88 (2011), 3163-3185. doi: 10.1080/00207160.2011.597503. [9] L. Zhang and Y. Xu, A full-newton step interior-point algorithm based on modified newton direction, Operations Research Letters, 39 (2011), 318-322. doi: 10.1016/j.orl.2011.06.006.
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