American Institute of Mathematical Sciences

Advanced
2013, 3(2): 223-234. doi: 10.3934/naco.2013.3.223

Subspace trust-region algorithm with conic model for unconstrained optimization

Xin Zhang 1, , Jie Wen 2, and  Qin Ni 2,

1. 

Jinling College of Nanjing University, Nanjing 210089, China
2. 

Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received  December 2011 Revised  January 2013 Published  April 2013

In this paper, a new subspace algorithm is proposed for unconstrained optimization. In this new algorithm, the subspace technique is used in the trust region subproblem with conic model, and the dogleg method is modified to solve this subproblem. The global convergence of this algorithm under some reasonable conditions is established. Numerical experiment shows that this algorithm may be superior to the corresponding algorithm without using subspace technique especially for large scale problems.
Keywords: conic model, trust region method, Unconstrained optimization, subspace method, global convergence..
Mathematics Subject Classification: Primary: 90C30; Secondary: 65K0.
Citation: Xin Zhang, Jie Wen, Qin Ni. Subspace trust-region algorithm with conic model for unconstrained optimization. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 223-234. doi: 10.3934/naco.2013.3.223
References:
[1]

W. C. Davidon, Conic approximation and collinear scaling for optimizers,, SIAM J.Number. Anal., 17 (1980), 268.  doi: 10.1137/0717023.  Google Scholar

[2]

S. Di and W. Y. Sun, A trust region Method for conic model to solve unconstrained optimization,, Optimization Methods and Software, 6 (1996), 237.  doi: 10.1080/10556789608805637.  Google Scholar

[3]

S. D. Jiang, "A Quasi-Newton Trust Region Method with a New Conic Model for the Unconstrained Optimization,", S. M thesis, (2005).   Google Scholar

[4]

X. P. Lu, Q. Ni and H. Liu, A dogleg method for solving new trust region subproblems of conic model,, Acta Math. Appl. Sin (in Chinese), 30 (2007), 855.   Google Scholar

[5]

X. P. Lu and Q. Ni, A quasi-Newton trust region method with a new conic model for the unconstrained optimization,, Applied Mathematics and Computation, 204 (2008), 373.  doi: 10.1016/j.amc.2008.06.062.  Google Scholar

[6]

J. J. More, B. S. Garbow and K. E. Hillstrom, Testing unconstrained optimization software,, ACM Trans. Math. Software, 7 (1981), 17.  doi: 10.1145/355934.355936.  Google Scholar

[7]

Q. Ni, Optimality conditions for trust-region subproblems involving a conic model,, SIAM J. Optimization, 15 (2005), 826.  doi: 10.1137/S1052623402418991.  Google Scholar

[8]

M. J. D. Powell, A new algorithm for unconstrained optimization,, in, (1970), 31.   Google Scholar

[9]

M. J. D. Powell, A hybrid method for nonlinear equations,, in, (1970), 87.   Google Scholar

[10]

M. J. D. Powell, Convergence properties of a class of minimization algorithms,, in, (1974), 1.   Google Scholar

[11]

M. J. D. Powell and Y. X. Yuan, A trust region algorithm for equality constrained optimization,, Math. Program., 49 (1991), 189.  doi: 10.1007/BF01588787.  Google Scholar

[12]

Y. X. Yuan, A review of trust region algorithms for optimization,, in, (2000), 271.   Google Scholar

[13]

H. W. Zhou and Y. X. Yuan, A subspace implementation of quasi-newton trust region methods for unconstrained optimization,, Report, (2004).   Google Scholar

[14]

M. F. Zhu, Y. Xue and F. S. Zhang, A quasi-Newton type trust region method based on the conic model,, Numer. Math. (A Journal of Chinese Universities), 17 (1995), 36.   Google Scholar

show all references

References:
[1]

W. C. Davidon, Conic approximation and collinear scaling for optimizers,, SIAM J.Number. Anal., 17 (1980), 268.  doi: 10.1137/0717023.  Google Scholar

[2]

S. Di and W. Y. Sun, A trust region Method for conic model to solve unconstrained optimization,, Optimization Methods and Software, 6 (1996), 237.  doi: 10.1080/10556789608805637.  Google Scholar

[3]

S. D. Jiang, "A Quasi-Newton Trust Region Method with a New Conic Model for the Unconstrained Optimization,", S. M thesis, (2005).   Google Scholar

[4]

X. P. Lu, Q. Ni and H. Liu, A dogleg method for solving new trust region subproblems of conic model,, Acta Math. Appl. Sin (in Chinese), 30 (2007), 855.   Google Scholar

[5]

X. P. Lu and Q. Ni, A quasi-Newton trust region method with a new conic model for the unconstrained optimization,, Applied Mathematics and Computation, 204 (2008), 373.  doi: 10.1016/j.amc.2008.06.062.  Google Scholar

[6]

J. J. More, B. S. Garbow and K. E. Hillstrom, Testing unconstrained optimization software,, ACM Trans. Math. Software, 7 (1981), 17.  doi: 10.1145/355934.355936.  Google Scholar

[7]

Q. Ni, Optimality conditions for trust-region subproblems involving a conic model,, SIAM J. Optimization, 15 (2005), 826.  doi: 10.1137/S1052623402418991.  Google Scholar

[8]

M. J. D. Powell, A new algorithm for unconstrained optimization,, in, (1970), 31.   Google Scholar

[9]

M. J. D. Powell, A hybrid method for nonlinear equations,, in, (1970), 87.   Google Scholar

[10]

M. J. D. Powell, Convergence properties of a class of minimization algorithms,, in, (1974), 1.   Google Scholar

[11]

M. J. D. Powell and Y. X. Yuan, A trust region algorithm for equality constrained optimization,, Math. Program., 49 (1991), 189.  doi: 10.1007/BF01588787.  Google Scholar

[12]

Y. X. Yuan, A review of trust region algorithms for optimization,, in, (2000), 271.   Google Scholar

[13]

H. W. Zhou and Y. X. Yuan, A subspace implementation of quasi-newton trust region methods for unconstrained optimization,, Report, (2004).   Google Scholar

[14]

M. F. Zhu, Y. Xue and F. S. Zhang, A quasi-Newton type trust region method based on the conic model,, Numer. Math. (A Journal of Chinese Universities), 17 (1995), 36.   Google Scholar

[1]

Lijuan Zhao, Wenyu Sun. Nonmonotone retrospective conic trust region method for unconstrained optimization. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 309-325. doi: 10.3934/naco.2013.3.309
[2]

Jun Chen, Wenyu Sun, Zhenghao Yang. A non-monotone retrospective trust-region method for unconstrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (4) : 919-944. doi: 10.3934/jimo.2013.9.919
[3]

Nobuko Sagara, Masao Fukushima. trust region method for nonsmooth convex optimization. Journal of Industrial & Management Optimization, 2005, 1 (2) : 171-180. doi: 10.3934/jimo.2005.1.171
[4]

Honglan Zhu, Qin Ni, Meilan Zeng. A quasi-Newton trust region method based on a new fractional model. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 237-249. doi: 10.3934/naco.2015.5.237
[5]

Liang Zhang, Wenyu Sun, Raimundo J. B. de Sampaio, Jinyun Yuan. A wedge trust region method with self-correcting geometry for derivative-free optimization. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 169-184. doi: 10.3934/naco.2015.5.169
[6]

Jun Takaki, Nobuo Yamashita. A derivative-free trust-region algorithm for unconstrained optimization with controlled error. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 117-145. doi: 10.3934/naco.2011.1.117
[7]

Dan Xue, Wenyu Sun, Hongjin He. A structured trust region method for nonconvex programming with separable structure. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 283-293. doi: 10.3934/naco.2013.3.283
[8]

Guanghui Zhou, Qin Ni, Meilan Zeng. A scaled conjugate gradient method with moving asymptotes for unconstrained optimization problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 595-608. doi: 10.3934/jimo.2016034
[9]

Chunlin Hao, Xinwei Liu. A trust-region filter-SQP method for mathematical programs with linear complementarity constraints. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1041-1055. doi: 10.3934/jimo.2011.7.1041
[10]

Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A socp relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019104
[11]

Bülent Karasözen. Survey of trust-region derivative free optimization methods. Journal of Industrial & Management Optimization, 2007, 3 (2) : 321-334. doi: 10.3934/jimo.2007.3.321
[12]

Chien-Wen Chao, Shu-Cherng Fang, Ching-Jong Liao. A tropical cyclone-based method for global optimization. Journal of Industrial & Management Optimization, 2012, 8 (1) : 103-115. doi: 10.3934/jimo.2012.8.103
[13]

Weijun Zhou, Youhua Zhou. On the strong convergence of a modified Hestenes-Stiefel method for nonconvex optimization. Journal of Industrial & Management Optimization, 2013, 9 (4) : 893-899. doi: 10.3934/jimo.2013.9.893
[14]

Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013
[15]

Liyan Qi, Xiantao Xiao, Liwei Zhang. On the global convergence of a parameter-adjusting Levenberg-Marquardt method. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 25-36. doi: 10.3934/naco.2015.5.25
[16]

Jing Zhou, Dejun Chen, Zhenbo Wang, Wenxun Xing. A conic approximation method for the 0-1 quadratic knapsack problem. Journal of Industrial & Management Optimization, 2013, 9 (3) : 531-547. doi: 10.3934/jimo.2013.9.531
[17]

Wen-ling Zhao, Dao-jin Song. A global error bound via the SQP method for constrained optimization problem. Journal of Industrial & Management Optimization, 2007, 3 (4) : 775-781. doi: 10.3934/jimo.2007.3.775
[18]

Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020139
[19]

Qiang Long, Changzhi Wu. A hybrid method combining genetic algorithm and Hooke-Jeeves method for constrained global optimization. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1279-1296. doi: 10.3934/jimo.2014.10.1279
[20]

Qiao Liang, Qiang Ye. Deflation by restriction for the inverse-free preconditioned Krylov subspace method. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 55-71. doi: 10.3934/naco.2016.6.55

 Impact Factor: 

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences