American Institute of Mathematical Sciences

Advanced
2015, 5(2): 211-231. doi: 10.3934/naco.2015.5.211

Primal-dual interior-point algorithms for convex quadratic circular cone optimization

Yanqin Bai 1, , Xuerui Gao 1, and  Guoqiang Wang 2,

1. 

Department of Mathematics, Shanghai University, Shanghai 200444, China, China
2. 

College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620

Received  December 2014 Revised  May 2015 Published  June 2015

In this paper we focus on a class of special nonsymmetric cone optimization problem called circular cone optimization problem, which has a convex quadratic function as the objective function and an intersection of a non-self-dual circular cone and linear equations as the constraint condition. Firstly we establish the algebraic relationships between the circular cone and the second-order cone and translate the original problem from the circular cone optimization problem to the second-order cone optimization problem. Then we present kernel-function based primal-dual interior-point algorithms for solving this special circular cone optimization and derive the iteration bounds for large- and small-update methods. Finally, some preliminary numerical results are provided to demonstrate the computational performance of the proposed algorithms.
Keywords: circular cone optimization, large- and small-update methods, second-order cone optimization, polynomial complexity., Interior-point methods, Kernel function.
Mathematics Subject Classification: Primary: 90C05, 90C51; Secondary: 65K0.
Citation: Yanqin Bai, Xuerui Gao, Guoqiang Wang. Primal-dual interior-point algorithms for convex quadratic circular cone optimization. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 211-231. doi: 10.3934/naco.2015.5.211
References:
[1]

Math. Program., 95 (2003), 3-51. doi: 10.1007/s10107-002-0339-5.  Google Scholar

[2]

Internat. Ser. Oper. Res. Management Sci., 166 (2012). doi: 10.1007/978-1-4614-0769-0.  Google Scholar

[3]

Science Press, Beijing, 2010. Google Scholar

[4]

SIAM J. Optim., 15 (2004), 101-128. doi: 10.1137/S1052623403423114.  Google Scholar

[5]

Nonlinear Anal., 70 (2009), 3584-3602. doi: 10.1016/j.na.2008.07.016.  Google Scholar

[6]

IEEE Trans. Robot., 23 (2007), 1117-1132. Google Scholar

[7]

Nonlinear Anal., 85 (2013), 160-173. doi: 10.1016/j.na.2013.01.017.  Google Scholar

[8]

Positivity, 1 (1997), 331-357. doi: 10.1023/A:1009701824047.  Google Scholar

[9]

European J. Oper. Res., 214(2011), 473-484. doi: 10.1016/j.ejor.2011.02.022.  Google Scholar

[10]

Jpn. J. Ind. Appl. Math., 29 (2012), 499-517. doi: 10.1007/s13160-012-0081-1.  Google Scholar

[11]

Optim. Method Softw., 27 (2012) 893-917. doi: 10.1080/10556788.2011.567270.  Google Scholar

[12]

SIAM J. Optim., 13 (2002), 179-203. doi: 10.1137/S1052623401383236.  Google Scholar

[13]

John Wiley & Sons, 1997.  Google Scholar

[14]

Math. Program., 96 (2003), 409-438. doi: 10.1007/s10107-003-0380-z.  Google Scholar

[15]

Math. Program., 150 (2015), 391-422. doi: 10.1007/s10107-014-0773-1.  Google Scholar

[16]

Appl. Math. Comput., 215 (2009), 1047-1061. doi: 10.1016/j.amc.2009.06.034.  Google Scholar

[17]

Nonlinear Anal., 71 (2009), 3389-3402. doi: 10.1016/j.na.2009.01.241.  Google Scholar

[18]

Numer. Algorithms, 57 (2011), 537-558. doi: 10.1007/s11075-010-9444-3.  Google Scholar

[19]

Linear Algebra Appl., 433 (2010), 718-736. doi: 10.1016/j.laa.2010.03.042.  Google Scholar

[20]

Proceedings of 2010 IEEE Multi-Conference on Systems and Control, (2010), 13-19. Google Scholar

[21]

J. Optim. Theory Appl., 158 (2013), 816-858. doi: 10.1007/s10957-013-0278-8.  Google Scholar

[22]

Abstr. Appl. Anal., 2014 (2014), 21pages. doi: 10.1155/2014/603542.  Google Scholar

[23]

J. Nonlinear Convex Anal., 14 (2013), 807-816.  Google Scholar

[24]

J. Inequal. Appl., 2013 (2013), 17 pages. doi: 10.1186/1029-242X-2013-571.  Google Scholar

[25]

Optim., 64 (2014), 113-147. doi: 10.1080/02331934.2014.951043.  Google Scholar

show all references

References:
[1]

Math. Program., 95 (2003), 3-51. doi: 10.1007/s10107-002-0339-5.  Google Scholar

[2]

Internat. Ser. Oper. Res. Management Sci., 166 (2012). doi: 10.1007/978-1-4614-0769-0.  Google Scholar

[3]

Science Press, Beijing, 2010. Google Scholar

[4]

SIAM J. Optim., 15 (2004), 101-128. doi: 10.1137/S1052623403423114.  Google Scholar

[5]

Nonlinear Anal., 70 (2009), 3584-3602. doi: 10.1016/j.na.2008.07.016.  Google Scholar

[6]

IEEE Trans. Robot., 23 (2007), 1117-1132. Google Scholar

[7]

Nonlinear Anal., 85 (2013), 160-173. doi: 10.1016/j.na.2013.01.017.  Google Scholar

[8]

Positivity, 1 (1997), 331-357. doi: 10.1023/A:1009701824047.  Google Scholar

[9]

European J. Oper. Res., 214(2011), 473-484. doi: 10.1016/j.ejor.2011.02.022.  Google Scholar

[10]

Jpn. J. Ind. Appl. Math., 29 (2012), 499-517. doi: 10.1007/s13160-012-0081-1.  Google Scholar

[11]

Optim. Method Softw., 27 (2012) 893-917. doi: 10.1080/10556788.2011.567270.  Google Scholar

[12]

SIAM J. Optim., 13 (2002), 179-203. doi: 10.1137/S1052623401383236.  Google Scholar

[13]

John Wiley & Sons, 1997.  Google Scholar

[14]

Math. Program., 96 (2003), 409-438. doi: 10.1007/s10107-003-0380-z.  Google Scholar

[15]

Math. Program., 150 (2015), 391-422. doi: 10.1007/s10107-014-0773-1.  Google Scholar

[16]

Appl. Math. Comput., 215 (2009), 1047-1061. doi: 10.1016/j.amc.2009.06.034.  Google Scholar

[17]

Nonlinear Anal., 71 (2009), 3389-3402. doi: 10.1016/j.na.2009.01.241.  Google Scholar

[18]

Numer. Algorithms, 57 (2011), 537-558. doi: 10.1007/s11075-010-9444-3.  Google Scholar

[19]

Linear Algebra Appl., 433 (2010), 718-736. doi: 10.1016/j.laa.2010.03.042.  Google Scholar

[20]

Proceedings of 2010 IEEE Multi-Conference on Systems and Control, (2010), 13-19. Google Scholar

[21]

J. Optim. Theory Appl., 158 (2013), 816-858. doi: 10.1007/s10957-013-0278-8.  Google Scholar

[22]

Abstr. Appl. Anal., 2014 (2014), 21pages. doi: 10.1155/2014/603542.  Google Scholar

[23]

J. Nonlinear Convex Anal., 14 (2013), 807-816.  Google Scholar

[24]

J. Inequal. Appl., 2013 (2013), 17 pages. doi: 10.1186/1029-242X-2013-571.  Google Scholar

[25]

Optim., 64 (2014), 113-147. doi: 10.1080/02331934.2014.951043.  Google Scholar

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