# 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 & 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. doi: 10.1007/s10957-009-9634-0. Google Scholar [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. doi: 10.1080/10556780600816692. Google Scholar [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: , (). Google Scholar [4] C. Roos, A full-Newton step $\mathcal O(n)$ infeasible interior-point algorithm for linear optimization,, SIAM Journal on Optimization, 16 (2006), 1110. doi: 10.1137/050623917. Google Scholar [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. doi: 10.1137/140975462. Google Scholar [6] C. Roos, T. Terlaky and J. P. Vial, Interior Point Methods for Linear Optimization,, Revised edition, (2006). Google Scholar [7] S. Wright, Primal-dual Interior-point Methods,, SIAM, (1997). doi: 10.1137/1.9781611971453. Google Scholar [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. doi: 10.1080/00207160.2011.597503. Google Scholar [9] L. Zhang and Y. Xu, A full-newton step interior-point algorithm based on modified newton direction,, Operations Research Letters, 39 (2011), 318. doi: 10.1016/j.orl.2011.06.006. Google Scholar

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. doi: 10.1007/s10957-009-9634-0. Google Scholar [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. doi: 10.1080/10556780600816692. Google Scholar [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: , (). Google Scholar [4] C. Roos, A full-Newton step $\mathcal O(n)$ infeasible interior-point algorithm for linear optimization,, SIAM Journal on Optimization, 16 (2006), 1110. doi: 10.1137/050623917. Google Scholar [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. doi: 10.1137/140975462. Google Scholar [6] C. Roos, T. Terlaky and J. P. Vial, Interior Point Methods for Linear Optimization,, Revised edition, (2006). Google Scholar [7] S. Wright, Primal-dual Interior-point Methods,, SIAM, (1997). doi: 10.1137/1.9781611971453. Google Scholar [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. doi: 10.1080/00207160.2011.597503. Google Scholar [9] L. Zhang and Y. Xu, A full-newton step interior-point algorithm based on modified newton direction,, Operations Research Letters, 39 (2011), 318. doi: 10.1016/j.orl.2011.06.006. Google Scholar
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