An infeasible full NT-step interior point method for circular optimization

Behrouz Kheirfam; Guoqiang Wang

doi:10.3934/naco.2017011

Article Contents

2017, Volume 7, Issue 2: 171-184. Doi: 10.3934/naco.2017011

This issue Previous Article Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems Next Article The optimal stabilization of orbital motion in a neighborhood of collinear libration point

An infeasible full NT-step interior point method for circular optimization

Behrouz Kheirfam^1, , and
Guoqiang Wang^2,

1.
Department of Applied Mathematics, Azarbaijan Shahid Madani University, Tabriz, I. R. Iran
2.
College of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai, 201620, China

^* Corresponding author: Behrouz Kheirfam
^* Corresponding author: Behrouz Kheirfam

Received: July 2016

Revised: May 2017

Published: October 2017

The second author was supported by National Natural Science Foundation of China (No. 11471211) and Shanghai Natural Science Fund Project (No. 14ZR1418900).

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Abstract

In this paper, we design a primal-dual infeasible interior-point method for circular optimization that uses only full Nesterov-Todd steps. Each main iteration of the algorithm consisted of one so-called feasibility step. Furthermore, giving a complexity analysis of the algorithm, we derive the currently best-known iteration bound for infeasible interior-point methods.

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Mathematics Subject Classification: 90C51.

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References

[1]	B. Alzalg, Primal-dual path-following algorithm for circular programming, Available from: http://www.optimization-online.org/DB_FILE/2015/07/4998.pdf.
[2]	B. Alzalg, The circular cone: A new paradigm for symmetric cone in a Euclidean Jordan algebra, Available from: http://www.optimization-online.org/DB_HTML/2015/07/5027.html.
[3]	Y. Bai, P. Ma and J. Zhang, A polynomial-time interior-point method for circular cone programming based on kernel functions, J. Ind. Manag. Optim., 12 (2015), 739-756. doi: 10.3934/jimo.2016.12.739.
[4]	X. N. Chi, H. J. Wei, Y. Q. Feng and J. W. Chen, Variational inequality formulation of circular cone eigenvalue complementarity problems, Fixed Point Theory Appl., 2016: 31 (2016). doi: 10.1186/s13663-016-0514-7.
[5]	L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, Positivity, 1 (1997), 331-357. doi: 10.1023/A:1009701824047.
[6]	B. Kheirfam, An interior-point method for Cartesian $P_(κ)$ -linear complementarity problem over symmetric cones, ORiON*, 30 (2014), 41-58.
[7]	B. Kheirfam, A full NT-step infeasible interior-point algorithm for semidefinite optimization based on a self-regular proximity, The ANZIAM J., 53 (2011), 48-67. doi: 10.1017/S144618111200003X.
[8]	B. Kheirfam, Simplified infeasible interior-point algorithm for SDO using full Nesterov-Todd step, Numer. Algorithms, 59 (2012), 589-606. doi: 10.1007/s11075-011-9506-1.
[9]	B. Kheirfam, A new complexity analysis for full-Newton step infeasible interior-point algorithm for horizontal linear complementarity problems, J. Optim. Theory Appl., 161 (2014), 853-869. doi: 10.1007/s10957-013-0457-7.
[10]	B. Kheirfam, A new infeasible interior-point method based on Darvay's technique for symmetric optimization}, Ann. Oper. Res., 211 (2013), 209-224. doi: 10.1007/s10479-013-1474-5.
[11]	B. Kheirfam, Simplified analysis of a full Nesterov-Todd step infeasible interior-point method for symmetric optimization, Asian-Eur. J. Math., 8 (2015), 1550071 (14 pages). doi: 10.1142/S1793557115500710.
[12]	B. Kheirfam, An improved full-Newton step $O(n)$ infeasible interior-point method for horizontal linear complementarity problem, Numer. Algorithms, 71 (2016), 491-503. doi: 10.1007/s11075-015-0005-7.
[13]	B. Kheirfam, A full step infeasible interior-point method for Cartesian $P_(κ)$ -SCLCP, Optim. Letters*, 10 (2016), 591-603. doi: 10.1007/s11590-015-0884-5.
[14]	B. Kheirfam, An improved and modified infeasible interior-point method for symmetric optimization, Asian-Eur. J. Math., 9 (2016), 1650059 (13 pages). doi: 10.1142/S1793557116500595.
[15]	B. Kheirfam, A full Nesterov-Todd step interior-point method for circular cone optimization, Commun. Comb. Optim., 1 (2016), 83-102.
[16]	B. Kheirfam and N. Mahdavi-Amiri, A new interior-point algorithm based on modified Nesterov-Todd direction for symmetric cone linear complementarity problem, Optim. Letters, 8 (2014), 1017-1029. doi: 10.1007/s11590-013-0618-5.
[17]	B. Kheirfam and N. Mahdavi-Amiri, A full Nesterov-Todd step infeasible interior-point algorithm for symmetric cone linear complementarity problem, Bull. Iranian Math. Soc., 40 (2014), 541-564.
[18]	X. Miao, S. Guo, N. Qi and J. S. Chen, Constructions of complementarity functions and merit functions for circular cone complementarity problem, Comput. Optim. Appl., 63 (2016), 495-522. doi: 10.1007/s10589-015-9781-1.
[19]	R. D. C. Monteiro and Y. Zhang, A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite programming, Math. Program., 81 (1998), 281-299. doi: 10.1007/BF01580085.
[20]	Y. E. Nesterov and M. J. Todd, Primal-dual interior-point methods for self-scaled cones, SIAM J. Optim., 8 (1998), 324-364. doi: 10.1137/S1052623495290209.
[21]	B. K. Rangarajan, Polynomial convergence of infeasible-interior-point methods over symmetric cones, SIAM J. Optim., 16 (2006), 1211-1229. doi: 10.1137/040606557.
[22]	C. Roos, A full-Newton step $O(n)$ infeasible interior-point algorithm for linear optimization, SIAM J. Optim., 16 (2006), 1110-1136. doi: 10.1137/050623917.
[23]	C. Roos, An improved and simplified full-Newton step $O(n)$ infeasible interior-point method for linear optimization, SIAM J. Optim., 25 (2015), 102-114. doi: 10.1137/140975462.
[24]	C. Roos, T. Terlaky and J-Ph. Vial, Theory and Algorithms for Linear Optimization. An Interior-Point Approach, John Wiley and Sons, Chichester, UK, 1997.
[25]	S. H. Schmieta and F. Alizadeh, Extension of primal-dual interior point methods to symmetric cones, Math. Program., 96 (2003), 409-438. doi: 10.1007/s10107-003-0380-z.
[26]	G. Q. Wang, L. C. Kong, J. Y. Tao and G. Lesaja, Improved complexity analysis of full Nesterov-Todd step feasible interior-point method for symmetric optimization, J. Optim. Theory Appl., 166 (2015), 588-604. doi: 10.1007/s10957-014-0696-2.
[27]	G. Q. Wang and G. Lesaja, Full Nesterov-Todd step feasible interior-point method for the Cartesian $P_(κ)$ -SCLCP, Optim. Methods & Softw.*, 28 (2013), 600-618. doi: 10.1080/10556788.2013.781600.
[28]	G. Q. Wang, C. J. Yu and K. L. Teo, A new full Nesterov-Todd step feasible interior-point method for convex quadratic symmetric cone optimization, Appl. Math. Comput., 221 (2013), 329-343. doi: 10.1016/j.amc.2013.06.064.
[29]	J. Zhou, J. S. Chen and B. S. Mordukhovich, Variational analysis of circular cone programs, Optimization, 64 (2014), 113-147. doi: 10.1080/02331934.2014.951043.