American Institute of Mathematical Sciences

2015, 5(1): 37-46. doi: 10.3934/naco.2015.5.37

Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization

 1 College of Mathematics and Physics, Bohai University, Jinzhou, MO 121000, China, China

Received  January 2015 Revised  March 2015 Published  March 2015

Kernel functions play an important role in defining new search directions for primal-dual interior-point algorithm. In this paper, a new kernel function which its barrier term is integral type is proposed. We study the properties of the new kernel function, and give a primal-dual interior-point algorithm for solving linear optimization based on the new kernel function. Polynomial complexity of algorithm is analyzed. The iteration bounds both for large-update and for small-update methods are obtained, respectively. The iteration bound for small-update method is the best known complexity bound.
Citation: Siqi Li, Weiyi Qian. Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 37-46. doi: 10.3934/naco.2015.5.37
References:
 [1] Y. Q. Bai, M. El Ghami and C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization, SIAM Journal on Optimization, 15 (2004), 101-128. doi: 10.1137/S1052623403423114.  Google Scholar [2] Y. Q. Bai, J. Guo and C. Roos, A new kernel function yielding the best known iteration bounds for primal-dual interior-point algorithms, Acta Mathematica Sinica English Series, 25 (2009), 2169-2178. doi: 10.1007/s10114-009-6457-8.  Google Scholar [3] Y. Q. Bai and C. Roos, A polynomial-time algorithm for linear optimization based on a new simple kernel function, Optimization Methods and Software, 18 (2003), 631-646. doi: 10.1080/10556780310001639735.  Google Scholar [4] Y. Q. Bai, M. El Ghami and C. Roos, A new efficient large-update primal-dual interior-point method based on a finite barrier, SIAM Journal on Optimization, 13 (2002), 766-782. doi: 10.1137/S1052623401398132.  Google Scholar [5] Y. Q. Bai, C. Roos and M. El Ghami, A primal-dual interior-point method for linear optimization based on a new proximity function, Optimization Methods and Software, 17 (2002), 985-1008. doi: 10.1080/1055678021000090024.  Google Scholar [6] N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica, 4 (1984), 373-395. doi: 10.1007/BF02579150.  Google Scholar [7] Y. Nesterov and A. Nemirovskii, Interior-point polynomial methods in convex programming, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970791.  Google Scholar [8] J. M. Peng, C. Roos and T. Terlaky, A new class of polynomial primal-dual methods for linear and semidefinite programming, European Journal of Operational Research, 143 (2002), 234-256. doi: 10.1016/S0377-2217(02)00275-8.  Google Scholar [9] J. Renegar, A polynomial time algorithm based on Newton's method for linear programming, Mathematical Programming, 40 (1988), 59-94. doi: 10.1007/BF01580724.  Google Scholar [10] C. Roos and J. P. Vial, A polynomial method of approximate centers for linear programming, Mathematical Programming, 54 (1992), 295-305. doi: 10.1007/BF01586056.  Google Scholar

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References:
 [1] Y. Q. Bai, M. El Ghami and C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization, SIAM Journal on Optimization, 15 (2004), 101-128. doi: 10.1137/S1052623403423114.  Google Scholar [2] Y. Q. Bai, J. Guo and C. Roos, A new kernel function yielding the best known iteration bounds for primal-dual interior-point algorithms, Acta Mathematica Sinica English Series, 25 (2009), 2169-2178. doi: 10.1007/s10114-009-6457-8.  Google Scholar [3] Y. Q. Bai and C. Roos, A polynomial-time algorithm for linear optimization based on a new simple kernel function, Optimization Methods and Software, 18 (2003), 631-646. doi: 10.1080/10556780310001639735.  Google Scholar [4] Y. Q. Bai, M. El Ghami and C. Roos, A new efficient large-update primal-dual interior-point method based on a finite barrier, SIAM Journal on Optimization, 13 (2002), 766-782. doi: 10.1137/S1052623401398132.  Google Scholar [5] Y. Q. Bai, C. Roos and M. El Ghami, A primal-dual interior-point method for linear optimization based on a new proximity function, Optimization Methods and Software, 17 (2002), 985-1008. doi: 10.1080/1055678021000090024.  Google Scholar [6] N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica, 4 (1984), 373-395. doi: 10.1007/BF02579150.  Google Scholar [7] Y. Nesterov and A. Nemirovskii, Interior-point polynomial methods in convex programming, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970791.  Google Scholar [8] J. M. Peng, C. Roos and T. Terlaky, A new class of polynomial primal-dual methods for linear and semidefinite programming, European Journal of Operational Research, 143 (2002), 234-256. doi: 10.1016/S0377-2217(02)00275-8.  Google Scholar [9] J. Renegar, A polynomial time algorithm based on Newton's method for linear programming, Mathematical Programming, 40 (1988), 59-94. doi: 10.1007/BF01580724.  Google Scholar [10] C. Roos and J. P. Vial, A polynomial method of approximate centers for linear programming, Mathematical Programming, 54 (1992), 295-305. doi: 10.1007/BF01586056.  Google Scholar
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