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

Yanqin  Bai; Xuerui  Gao; Guoqiang  Wang

doi:10.3934/naco.2015.5.211

Article Contents

2015, Volume 5, Issue 2: 211-231. Doi: 10.3934/naco.2015.5.211

This issue Previous Article Continuity and stability of two-stage stochastic programs with quadratic continuous recourse Next Article

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

1.
Department of Mathematics, Shanghai University, Shanghai 200444
2.
College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620

Received: December 2014

Revised: May 2015

Published: June 2015

Abstract / Introduction Related Papers Cited by

Abstract

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.

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Mathematics Subject Classification: Primary: 90C05, 90C51; Secondary: 65K05.

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References

[1]	F. Alizadeh and D. Goldfard, Second-order cone programming, Math. Program., 95 (2003), 3-51.doi: 10.1007/s10107-002-0339-5.
[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.
[3]	Y. Q. Bai, Kernel Function-Based Interior-Point Algorithms for Conic Optimization, Science Press, Beijing, 2010.
[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), 101-128.doi: 10.1137/S1052623403423114.
[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), 3584-3602.doi: 10.1016/j.na.2008.07.016.
[6]	S. P. Boyd and B. Wegbreit, Fast computation of optimal contact forces, IEEE Trans. Robot., 23 (2007), 1117-1132.
[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), 160-173.doi: 10.1016/j.na.2013.01.017.
[8]	L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, Positivity, 1 (1997), 331-357.doi: 10.1023/A:1009701824047.
[9]	G. Gu, M. Zangiabadi and C. Roos, Full Nesterov-Todd step interior-point method for symmetric optimization, European J. Oper. Res., 214(2011), 473-484.doi: 10.1016/j.ejor.2011.02.022.
[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), 499-517.doi: 10.1007/s13160-012-0081-1.
[11]	Y. Nesterov, Towards non-symmetric conic optimization, Optim. Method Softw., 27 (2012) 893-917.doi: 10.1080/10556788.2011.567270.
[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), 179-203.doi: 10.1137/S1052623401383236.
[13]	C. Roos, T. Terlaky and J.-Ph. Vial, Theory and Algorithms for Linear Optimization: An Interior-Point Approach, John Wiley & Sons, 1997.
[14]	S. H. Schmieta and F. Alizadeh, Extension of primal-dual interior-point algorithms to symmetric cones, Math. Program., 96 (2003), 409-438.doi: 10.1007/s10107-003-0380-z.
[15]	A. Skajaa and Y. Y. Ye, A homogeneous interior-point algorithm for nonsymmetric convex conic optimization, Math. Program., 150 (2015), 391-422.doi: 10.1007/s10107-014-0773-1.
[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), 1047-1061.doi: 10.1016/j.amc.2009.06.034.
[17]	G. Q. Wang and Y. Q. Bai, Primal-dual interior-point algorithm for convex quadratic semidefinite optimization, Nonlinear Anal., 71 (2009), 3389-3402.doi: 10.1016/j.na.2009.01.241.
[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), 537-558.doi: 10.1007/s11075-010-9444-3.
[19]	Y. N. Wang, N. H. Xiu and J. Y. Han, On cone of nonsymmetric positive semidefinite matrices, Linear Algebra Appl., 433 (2010), 718-736.doi: 10.1016/j.laa.2010.03.042.
[20]	A. Yoshise and Y. Matsukawa, On optimization over the doubly nonnegative cone, Proceedings of 2010 IEEE Multi-Conference on Systems and Control, (2010), 13-19.
[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), 816-858.doi: 10.1007/s10957-013-0278-8.
[22]	J. C. Zhou and J. S. Chen, The vector-valued functions associated with circular cones, Abstr. Appl. Anal., 2014 (2014), 21pages.doi: 10.1155/2014/603542.
[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), 807-816.
[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), 17 pages.doi: 10.1186/1029-242X-2013-571.
[25]	J. C. Zhou, J. S. Chen and B. S. Mordukhovich, Variational analysis of circular cone programs, Optim., 64 (2014), 113-147.doi: 10.1080/02331934.2014.951043.