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

[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

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

[1]

Guoqiang Wang, Zhongchen Wu, Zhongtuan Zheng, Xinzhong Cai. Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function with a trigonometric barrier term. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 101-113. doi: 10.3934/naco.2015.5.101
[2]

Yanqin Bai, Pengfei Ma, Jing Zhang. A polynomial-time interior-point method for circular cone programming based on kernel functions. Journal of Industrial & Management Optimization, 2016, 12 (2) : 739-756. doi: 10.3934/jimo.2016.12.739
[3]

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
[4]

Yanqin Bai, Lipu Zhang. A full-Newton step interior-point algorithm for symmetric cone convex quadratic optimization. Journal of Industrial & Management Optimization, 2011, 7 (4) : 891-906. doi: 10.3934/jimo.2011.7.891
[5]

Behrouz Kheirfam. A full Nesterov-Todd step infeasible interior-point algorithm for symmetric optimization based on a specific kernel function. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 601-614. doi: 10.3934/naco.2013.3.601
[6]

Liwei Zhang, Jihong Zhang, Yule Zhang. Second-order optimality conditions for cone constrained multi-objective optimization. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1041-1054. doi: 10.3934/jimo.2017089
[7]

Anurag Jayswala, Tadeusz Antczakb, Shalini Jha. Second order modified objective function method for twice differentiable vector optimization problems over cone constraints. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 133-145. doi: 10.3934/naco.2019010
[8]

Yi Zhang, Yong Jiang, Liwei Zhang, Jiangzhong Zhang. A perturbation approach for an inverse linear second-order cone programming. Journal of Industrial & Management Optimization, 2013, 9 (1) : 171-189. doi: 10.3934/jimo.2013.9.171
[9]

Behrouz Kheirfam, Morteza Moslemi. On the extension of an arc-search interior-point algorithm for semidefinite optimization. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 261-275. doi: 10.3934/naco.2018015
[10]

Behrouz Kheirfam, Guoqiang Wang. An infeasible full NT-step interior point method for circular optimization. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 171-184. doi: 10.3934/naco.2017011
[11]

Xiaoni Chi, Zhongping Wan, Zijun Hao. Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1111-1125. doi: 10.3934/jimo.2015.11.1111
[12]

Xiaoling Guo, Zhibin Deng, Shu-Cherng Fang, Wenxun Xing. Quadratic optimization over one first-order cone. Journal of Industrial & Management Optimization, 2014, 10 (3) : 945-963. doi: 10.3934/jimo.2014.10.945
[13]

Ye Tian, Qingwei Jin, Zhibin Deng. Quadratic optimization over a polyhedral cone. Journal of Industrial & Management Optimization, 2016, 12 (1) : 269-283. doi: 10.3934/jimo.2016.12.269
[14]

Ye Tian, Shu-Cherng Fang, Zhibin Deng, Wenxun Xing. Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming. Journal of Industrial & Management Optimization, 2013, 9 (3) : 703-721. doi: 10.3934/jimo.2013.9.703
[15]

Xi-De Zhu, Li-Ping Pang, Gui-Hua Lin. Two approaches for solving mathematical programs with second-order cone complementarity constraints. Journal of Industrial & Management Optimization, 2015, 11 (3) : 951-968. doi: 10.3934/jimo.2015.11.951
[16]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A smoothing Newton method for generalized Nash equilibrium problems with second-order cone constraints. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 1-18. doi: 10.3934/naco.2012.2.1
[17]

Li Chu, Bo Wang, Jie Zhang, Hong-Wei Zhang. Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020050
[18]

Boshi Tian, Xiaoqi Yang, Kaiwen Meng. An interior-point $l_{\frac{1}{2}}$-penalty method for inequality constrained nonlinear optimization. Journal of Industrial & Management Optimization, 2016, 12 (3) : 949-973. doi: 10.3934/jimo.2016.12.949
[19]

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
[20]

Jae-Hong Pyo, Jie Shen. Normal mode analysis of second-order projection methods for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 817-840. doi: 10.3934/dcdsb.2005.5.817

 Impact Factor: 

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences