American Institute of Mathematical Sciences

Advanced
October  2011, 7(4): 891-906. doi: 10.3934/jimo.2011.7.891

A full-Newton step interior-point algorithm for symmetric cone convex quadratic optimization

Yanqin Bai 1, and  Lipu Zhang 2,

1. 

Department of Mathematics, Shanghai University, 99, Shangda Road, 200444, Shanghai
2. 

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received  November 2009 Revised  May 2011 Published  August 2011

In this paper, we present a full-Newton step primal-dual interior-point algorithm for solving symmetric cone convex quadratic optimization problem, where the objective function is a convex quadratic function and the feasible set is the intersection of an affine subspace and a symmetric cone lies in Euclidean Jordan algebra. The search directions of the algorithm are obtained from the modification of NT-search direction in terms of the quadratic representation in Euclidean Jordan algebra. We prove that the algorithm has a quadratical convergence result. Furthermore, we present the complexity analysis for the algorithm and obtain the complexity bound as $\left\lceil 2\sqrt{r}\log\frac{\mu^0 r}{\epsilon}\right\rceil$, where $r$ is the rank of Euclidean Jordan algebras where the symmetric cone lies in.
Keywords: complexity analysis., Euclidean Jordan algebras, full-Newton step, NT-search direction, symmetric cone.
Mathematics Subject Classification: 17C99, 90C25, 90C5.
Citation: 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
References:
[1]

K. M. Anstreicher, D. den Hertog, C. Roos and T. Terlaky, A long-step barrier method for convex quadratic programming,, Algorithmica, 10 (1993), 365.  doi: 10.1007/BF01769704.  Google Scholar

[2]

Y. Q. Bai, M. El Ghami and C. Roos, A comparative study of kernel function for primal-dual interior-point algorithms in linear optimization,, SIAM J. Optim., 15 (2004), 101.  doi: 10.1137/S1052623403423114.  Google Scholar

[3]

S. Boyd and L. Vandenberghe, "Convex Optimization,", Cambridge University Press, (2004).   Google Scholar

[4]

J. Faraut and A. Korányi, "Analysis on Symmetric Cones,", Oxford Mathematical Monographs, (1994).   Google Scholar

[5]

L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms,, Special issue dedicated to William B. Gragg (Monterey, 86 (1997), 149.  doi: 10.1016/S0377-0427(97)00153-2.  Google Scholar

[6]

L. Faybusovich, A Jordan-algebraic approach to potential-reduction algorithms,, Math. Z., 239 (2002), 117.  doi: 10.1007/s002090100286.  Google Scholar

[7]

R. D. C. Monteiro and I. Adler, Interior path following primal-dual algorithms, II: Convex quadratic programming,, Math. Program., 44 (1989), 43.   Google Scholar

[8]

Y. E. Nesterov and M. J. Todd, Self-scaled barriers and interior-point methods for convex programming,, Math. Oper. Res., 22 (1997), 1.  doi: 10.1287/moor.22.1.1.  Google Scholar

[9]

J. Peng, C. Roos and T. Terlaky, "Self-regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms,", Princeton Series in Applied Mathematics, (2002).   Google Scholar

[10]

C. Roos, T. Terlaky and J.-Ph. Vial, "Theory and Algorithms for Linear Optimization. An Interior Point Approach,", Wiley-Interscience Series in Discrete Mathematics and Optimization, (1997).   Google Scholar

[11]

S. H. Schmieta and F. Alizadeh, Extension of primal-dual interior point algorithms to symmetric cones,, Math. Program, 96 (2003), 409.   Google Scholar

[12]

Changjun Yu, Kok Lay Teo, Liangsheng Zhang and Yanqin Bai, A new exact penalty function method for continuous inequality constrained optimization problems,, Journal of Industrial and Management Optimization, 4 (2010), 895.   Google Scholar

[13]

M. V. C. Vieira, "Jordan Algebraic Approach to Symmetric Optimization,", Ph.D thesis, (2007).   Google Scholar

show all references

References:
[1]

K. M. Anstreicher, D. den Hertog, C. Roos and T. Terlaky, A long-step barrier method for convex quadratic programming,, Algorithmica, 10 (1993), 365.  doi: 10.1007/BF01769704.  Google Scholar

[2]

Y. Q. Bai, M. El Ghami and C. Roos, A comparative study of kernel function for primal-dual interior-point algorithms in linear optimization,, SIAM J. Optim., 15 (2004), 101.  doi: 10.1137/S1052623403423114.  Google Scholar

[3]

S. Boyd and L. Vandenberghe, "Convex Optimization,", Cambridge University Press, (2004).   Google Scholar

[4]

J. Faraut and A. Korányi, "Analysis on Symmetric Cones,", Oxford Mathematical Monographs, (1994).   Google Scholar

[5]

L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms,, Special issue dedicated to William B. Gragg (Monterey, 86 (1997), 149.  doi: 10.1016/S0377-0427(97)00153-2.  Google Scholar

[6]

L. Faybusovich, A Jordan-algebraic approach to potential-reduction algorithms,, Math. Z., 239 (2002), 117.  doi: 10.1007/s002090100286.  Google Scholar

[7]

R. D. C. Monteiro and I. Adler, Interior path following primal-dual algorithms, II: Convex quadratic programming,, Math. Program., 44 (1989), 43.   Google Scholar

[8]

Y. E. Nesterov and M. J. Todd, Self-scaled barriers and interior-point methods for convex programming,, Math. Oper. Res., 22 (1997), 1.  doi: 10.1287/moor.22.1.1.  Google Scholar

[9]

J. Peng, C. Roos and T. Terlaky, "Self-regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms,", Princeton Series in Applied Mathematics, (2002).   Google Scholar

[10]

C. Roos, T. Terlaky and J.-Ph. Vial, "Theory and Algorithms for Linear Optimization. An Interior Point Approach,", Wiley-Interscience Series in Discrete Mathematics and Optimization, (1997).   Google Scholar

[11]

S. H. Schmieta and F. Alizadeh, Extension of primal-dual interior point algorithms to symmetric cones,, Math. Program, 96 (2003), 409.   Google Scholar

[12]

Changjun Yu, Kok Lay Teo, Liangsheng Zhang and Yanqin Bai, A new exact penalty function method for continuous inequality constrained optimization problems,, Journal of Industrial and Management Optimization, 4 (2010), 895.   Google Scholar

[13]

M. V. C. Vieira, "Jordan Algebraic Approach to Symmetric Optimization,", Ph.D thesis, (2007).   Google Scholar

[1]

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
[2]

Yu-Lin Chang, Chin-Yu Yang. Some useful inequalities via trace function method in Euclidean Jordan algebras. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 39-48. doi: 10.3934/naco.2014.4.39
[3]

Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial & Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733
[4]

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
[5]

Li-Xia Liu, Sanyang Liu, Chun-Feng Wang. Smoothing Newton methods for symmetric cone linear complementarity problem with the Cartesian $P$/$P_0$-property. Journal of Industrial & Management Optimization, 2011, 7 (1) : 53-66. doi: 10.3934/jimo.2011.7.53
[6]

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
[7]

Golamreza Zamani Eskandani, Hamid Vaezi. Hyers--Ulam--Rassias stability of derivations in proper Jordan $CQ^{*}$-algebras. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1469-1477. doi: 10.3934/dcds.2011.31.1469
[8]

M. P. de Oliveira. On 3-graded Lie algebras, Jordan pairs and the canonical kernel function. Electronic Research Announcements, 2003, 9: 142-151.

[9]

Xin-He Miao, Jein-Shan Chen. Error bounds for symmetric cone complementarity problems. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 627-641. doi: 10.3934/naco.2013.3.627
[10]

Yuan Shen, Lei Ji. Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step. Journal of Industrial & Management Optimization, 2019, 15 (1) : 159-175. doi: 10.3934/jimo.2018037
[11]

Peng Guo, Wenming Cheng, Yi Wang. A general variable neighborhood search for single-machine total tardiness scheduling problem with step-deteriorating jobs. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1071-1090. doi: 10.3934/jimo.2014.10.1071
[12]

Behrouz Kheirfam. A weighted-path-following method for symmetric cone linear complementarity problems. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 141-150. doi: 10.3934/naco.2014.4.141
[13]

Yi Zhang, Liwei Zhang, Jia Wu. On the convergence properties of a smoothing approach for mathematical programs with symmetric cone complementarity constraints. Journal of Industrial & Management Optimization, 2018, 14 (3) : 981-1005. doi: 10.3934/jimo.2017086
[14]

Liping Tang, Xinmin Yang, Ying Gao. Higher-order symmetric duality for multiobjective programming with cone constraints. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019033
[15]

Matthias Gerdts, Stefan Horn, Sven-Joachim Kimmerle. Line search globalization of a semismooth Newton method for operator equations in Hilbert spaces with applications in optimal control. Journal of Industrial & Management Optimization, 2017, 13 (1) : 47-62. doi: 10.3934/jimo.2016003
[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]

Leong-Kwan Li, Sally Shao. Convergence analysis of the weighted state space search algorithm for recurrent neural networks. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 193-207. doi: 10.3934/naco.2014.4.193
[18]

Ming-Jong Yao, Yu-Chun Wang. Theoretical analysis and a search procedure for the joint replenishment problem with deteriorating products. Journal of Industrial & Management Optimization, 2005, 1 (3) : 359-375. doi: 10.3934/jimo.2005.1.359
[19]

Juan Pablo Cárdenas, Gerardo Vidal, Gastón Olivares. Complexity, selectivity and asymmetry in the conformation of the power phenomenon. Analysis of Chilean society. Networks & Heterogeneous Media, 2015, 10 (1) : 167-194. doi: 10.3934/nhm.2015.10.167
[20]

Ai-Li Yang, Yu-Jiang Wu. Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 839-853. doi: 10.3934/naco.2012.2.839

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences