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]

F. Alizadeh and D. Goldfard, Second-order cone programming,, Math. Program., 95 (2003), 3.  doi: 10.1007/s10107-002-0339-5.  Google Scholar

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M. F. Anjos and J. B. Lasserre, Handbook on Semidefinite, Conic and Polynomial Optimization: Theory, Algorithms, Software and Applications,, Internat. Ser. Oper. Res. Management Sci., 166 (2012).  doi: 10.1007/978-1-4614-0769-0.  Google Scholar

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

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Y. Q. Bai, G. Q. Wang and C. Roos, Primal-dual interior-point algorithms for second-order cone optimization based on kernel functions,, Nonlinear Anal., 70 (2009), 3584.  doi: 10.1016/j.na.2008.07.016.  Google Scholar

[6]

S. P. Boyd and B. Wegbreit, Fast computation of optimal contact forces,, IEEE Trans. Robot., 23 (2007), 1117.   Google Scholar

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Y. L. Chang, C. Y. Yang and J. S. Chen, Smooth and nonsmooth analysis of vector-valued functions associated with circular cones,, Nonlinear Anal., 85 (2013), 160.  doi: 10.1016/j.na.2013.01.017.  Google Scholar

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L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms,, Positivity, 1 (1997), 331.  doi: 10.1023/A:1009701824047.  Google Scholar

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G. Gu, M. Zangiabadi and C. Roos, Full Nesterov-Todd step interior-point method for symmetric optimization,, European J. Oper. Res., 214 (2011), 473.  doi: 10.1016/j.ejor.2011.02.022.  Google Scholar

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Y. Matsukawa and A. Yoshise, A primal barrier function Phase I algorithm for nonsymmetric conic optimization problems,, Jpn. J. Ind. Appl. Math., 29 (2012), 499.  doi: 10.1007/s13160-012-0081-1.  Google Scholar

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Y. Nesterov, Towards non-symmetric conic optimization,, Optim. Method Softw., 27 (2012), 893.  doi: 10.1080/10556788.2011.567270.  Google Scholar

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J. Peng, C. Roos and T. Terlaky., A new class of polynomial primal-dual interior-point methods for second-order cone optimization based on self-regular proximities,, SIAM J. Optim., 13 (2002), 179.  doi: 10.1137/S1052623401383236.  Google Scholar

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C. Roos, T. Terlaky and J.-Ph. Vial, Theory and Algorithms for Linear Optimization: An Interior-Point Approach,, John Wiley & Sons, (1997).   Google Scholar

[14]

S. H. Schmieta and F. Alizadeh, Extension of primal-dual interior-point algorithms to symmetric cones,, Math. Program., 96 (2003), 409.  doi: 10.1007/s10107-003-0380-z.  Google Scholar

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A. Skajaa and Y. Y. Ye, A homogeneous interior-point algorithm for nonsymmetric convex conic optimization,, Math. Program., 150 (2015), 391.  doi: 10.1007/s10107-014-0773-1.  Google Scholar

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G. Q. Wang and Y. Q. Bai, A primal-dual path-following interior-point algorithm for second-order cone optimization with full Nesterov-Todd step,, Appl. Math. Comput., 215 (2009), 1047.  doi: 10.1016/j.amc.2009.06.034.  Google Scholar

[17]

G. Q. Wang and Y. Q. Bai, Primal-dual interior-point algorithm for convex quadratic semidefinite optimization,, Nonlinear Anal., 71 (2009), 3389.  doi: 10.1016/j.na.2009.01.241.  Google Scholar

[18]

G. Q. Wang and D. T. Zhu, A unified kernel function approach to primal-dual interior-point algorithms for convex quadratic SDO,, Numer. Algorithms, 57 (2011), 537.  doi: 10.1007/s11075-010-9444-3.  Google Scholar

[19]

Y. N. Wang, N. H. Xiu and J. Y. Han, On cone of nonsymmetric positive semidefinite matrices,, Linear Algebra Appl., 433 (2010), 718.  doi: 10.1016/j.laa.2010.03.042.  Google Scholar

[20]

A. Yoshise and Y. Matsukawa, On optimization over the doubly nonnegative cone,, Proceedings of 2010 IEEE Multi-Conference on Systems and Control, (2010), 13.   Google Scholar

[21]

M. Zangiabadi, G. Gu and C. Roos, A full Nesterov-Todd step infeasible interior-point method for second-order cone optimization,, J. Optim. Theory Appl., 158 (2013), 816.  doi: 10.1007/s10957-013-0278-8.  Google Scholar

[22]

J. C. Zhou and J. S. Chen, The vector-valued functions associated with circular cones,, Abstr. Appl. Anal., 2014 (2014).  doi: 10.1155/2014/603542.  Google Scholar

[23]

J. C. Zhou and J. S. Chen, Properties of circular cone and spectral factorization associated with circular cone,, J. Nonlinear Convex Anal., 14 (2013), 807.   Google Scholar

[24]

J. C. Zhou, J. S. Chen and H. F. Hung, Circular cone convexity and some inequalities associated with circular cones,, J. Inequal. Appl., 2013 (2013).  doi: 10.1186/1029-242X-2013-571.  Google Scholar

[25]

J. C. Zhou, J. S. Chen and B. S. Mordukhovich, Variational analysis of circular cone programs,, Optim., 64 (2014), 113.  doi: 10.1080/02331934.2014.951043.  Google Scholar

show all references

References:
[1]

F. Alizadeh and D. Goldfard, Second-order cone programming,, Math. Program., 95 (2003), 3.  doi: 10.1007/s10107-002-0339-5.  Google Scholar

[2]

M. F. Anjos and J. B. Lasserre, Handbook on Semidefinite, Conic and Polynomial Optimization: Theory, Algorithms, Software and Applications,, Internat. Ser. Oper. Res. Management Sci., 166 (2012).  doi: 10.1007/978-1-4614-0769-0.  Google Scholar

[3]

Y. Q. Bai, Kernel Function-Based Interior-Point Algorithms for Conic Optimization,, Science Press, (2010).   Google Scholar

[4]

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

[5]

Y. Q. Bai, G. Q. Wang and C. Roos, Primal-dual interior-point algorithms for second-order cone optimization based on kernel functions,, Nonlinear Anal., 70 (2009), 3584.  doi: 10.1016/j.na.2008.07.016.  Google Scholar

[6]

S. P. Boyd and B. Wegbreit, Fast computation of optimal contact forces,, IEEE Trans. Robot., 23 (2007), 1117.   Google Scholar

[7]

Y. L. Chang, C. Y. Yang and J. S. Chen, Smooth and nonsmooth analysis of vector-valued functions associated with circular cones,, Nonlinear Anal., 85 (2013), 160.  doi: 10.1016/j.na.2013.01.017.  Google Scholar

[8]

L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms,, Positivity, 1 (1997), 331.  doi: 10.1023/A:1009701824047.  Google Scholar

[9]

G. Gu, M. Zangiabadi and C. Roos, Full Nesterov-Todd step interior-point method for symmetric optimization,, European J. Oper. Res., 214 (2011), 473.  doi: 10.1016/j.ejor.2011.02.022.  Google Scholar

[10]

Y. Matsukawa and A. Yoshise, A primal barrier function Phase I algorithm for nonsymmetric conic optimization problems,, Jpn. J. Ind. Appl. Math., 29 (2012), 499.  doi: 10.1007/s13160-012-0081-1.  Google Scholar

[11]

Y. Nesterov, Towards non-symmetric conic optimization,, Optim. Method Softw., 27 (2012), 893.  doi: 10.1080/10556788.2011.567270.  Google Scholar

[12]

J. Peng, C. Roos and T. Terlaky., A new class of polynomial primal-dual interior-point methods for second-order cone optimization based on self-regular proximities,, SIAM J. Optim., 13 (2002), 179.  doi: 10.1137/S1052623401383236.  Google Scholar

[13]

C. Roos, T. Terlaky and J.-Ph. Vial, Theory and Algorithms for Linear Optimization: An Interior-Point Approach,, John Wiley & Sons, (1997).   Google Scholar

[14]

S. H. Schmieta and F. Alizadeh, Extension of primal-dual interior-point algorithms to symmetric cones,, Math. Program., 96 (2003), 409.  doi: 10.1007/s10107-003-0380-z.  Google Scholar

[15]

A. Skajaa and Y. Y. Ye, A homogeneous interior-point algorithm for nonsymmetric convex conic optimization,, Math. Program., 150 (2015), 391.  doi: 10.1007/s10107-014-0773-1.  Google Scholar

[16]

G. Q. Wang and Y. Q. Bai, A primal-dual path-following interior-point algorithm for second-order cone optimization with full Nesterov-Todd step,, Appl. Math. Comput., 215 (2009), 1047.  doi: 10.1016/j.amc.2009.06.034.  Google Scholar

[17]

G. Q. Wang and Y. Q. Bai, Primal-dual interior-point algorithm for convex quadratic semidefinite optimization,, Nonlinear Anal., 71 (2009), 3389.  doi: 10.1016/j.na.2009.01.241.  Google Scholar

[18]

G. Q. Wang and D. T. Zhu, A unified kernel function approach to primal-dual interior-point algorithms for convex quadratic SDO,, Numer. Algorithms, 57 (2011), 537.  doi: 10.1007/s11075-010-9444-3.  Google Scholar

[19]

Y. N. Wang, N. H. Xiu and J. Y. Han, On cone of nonsymmetric positive semidefinite matrices,, Linear Algebra Appl., 433 (2010), 718.  doi: 10.1016/j.laa.2010.03.042.  Google Scholar

[20]

A. Yoshise and Y. Matsukawa, On optimization over the doubly nonnegative cone,, Proceedings of 2010 IEEE Multi-Conference on Systems and Control, (2010), 13.   Google Scholar

[21]

M. Zangiabadi, G. Gu and C. Roos, A full Nesterov-Todd step infeasible interior-point method for second-order cone optimization,, J. Optim. Theory Appl., 158 (2013), 816.  doi: 10.1007/s10957-013-0278-8.  Google Scholar

[22]

J. C. Zhou and J. S. Chen, The vector-valued functions associated with circular cones,, Abstr. Appl. Anal., 2014 (2014).  doi: 10.1155/2014/603542.  Google Scholar

[23]

J. C. Zhou and J. S. Chen, Properties of circular cone and spectral factorization associated with circular cone,, J. Nonlinear Convex Anal., 14 (2013), 807.   Google Scholar

[24]

J. C. Zhou, J. S. Chen and H. F. Hung, Circular cone convexity and some inequalities associated with circular cones,, J. Inequal. Appl., 2013 (2013).  doi: 10.1186/1029-242X-2013-571.  Google Scholar

[25]

J. C. Zhou, J. S. Chen and B. S. Mordukhovich, Variational analysis of circular cone programs,, Optim., 64 (2014), 113.  doi: 10.1080/02331934.2014.951043.  Google Scholar

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