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 and 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), 268-281. doi: 10.1137/0717023.

[2]

S. Di and W. Y. Sun, A trust region Method for conic model to solve unconstrained optimization, Optimization Methods and Software, 6 (1996), 237-263. doi: 10.1080/10556789608805637.

[3]

S. D. Jiang, "A Quasi-Newton Trust Region Method with a New Conic Model for the Unconstrained Optimization," S. M thesis, Nanjing University of Aeronautics and Astronautics in Nanjing (in Chinese), 2005.

[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), 855-871.

[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), 373-384. doi: 10.1016/j.amc.2008.06.062.

[6]

J. J. More, B. S. Garbow and K. E. Hillstrom, Testing unconstrained optimization software, ACM Trans. Math. Software, 7 (1981), 17-41. doi: 10.1145/355934.355936.

[7]

Q. Ni, Optimality conditions for trust-region subproblems involving a conic model, SIAM J. Optimization, 15 (2005), 826-837. doi: 10.1137/S1052623402418991.

[8]

M. J. D. Powell, A new algorithm for unconstrained optimization, in "Nonlinear Programming" (eds. J. B. Rosen, O. L. Mangasarian and K. Ritter), Academic Press, New York, (1970), 31-66.

[9]

M. J. D. Powell, A hybrid method for nonlinear equations, in "Numerical Methods for Nonlinear Algebraic Equations " (eds. P. Robinowitz), Gordon and Breach Science, London, (1970), 87-144.

[10]

M. J. D. Powell, Convergence properties of a class of minimization algorithms, in "Nonlinear Programming 2" (eds. O. L. Mangasarian, R. R. Meyer and S. M. Robinson), Academic Press, New York, (1974), 1-27.

[11]

M. J. D. Powell and Y. X. Yuan, A trust region algorithm for equality constrained optimization, Math. Program., 49 (1991), 189-211. doi: 10.1007/BF01588787.

[12]

Y. X. Yuan, A review of trust region algorithms for optimization, in "ICIAM99: Proceedings of the Fourth International Congress on Industrial and Applied Mathematics" (eds. J. M. Ball and J. C. R. Hunt), Oxford University Press, Oxford UK, (2000), 271-282.

[13]

H. W. Zhou and Y. X. Yuan, A subspace implementation of quasi-newton trust region methods for unconstrained optimization, Report, ICMSEC, AMSS, Chinese Academy of Science, 2004.

[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), 36-47.

show all references

References:
[1]

W. C. Davidon, Conic approximation and collinear scaling for optimizers, SIAM J.Number. Anal., 17 (1980), 268-281. doi: 10.1137/0717023.

[2]

S. Di and W. Y. Sun, A trust region Method for conic model to solve unconstrained optimization, Optimization Methods and Software, 6 (1996), 237-263. doi: 10.1080/10556789608805637.

[3]

S. D. Jiang, "A Quasi-Newton Trust Region Method with a New Conic Model for the Unconstrained Optimization," S. M thesis, Nanjing University of Aeronautics and Astronautics in Nanjing (in Chinese), 2005.

[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), 855-871.

[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), 373-384. doi: 10.1016/j.amc.2008.06.062.

[6]

J. J. More, B. S. Garbow and K. E. Hillstrom, Testing unconstrained optimization software, ACM Trans. Math. Software, 7 (1981), 17-41. doi: 10.1145/355934.355936.

[7]

Q. Ni, Optimality conditions for trust-region subproblems involving a conic model, SIAM J. Optimization, 15 (2005), 826-837. doi: 10.1137/S1052623402418991.

[8]

M. J. D. Powell, A new algorithm for unconstrained optimization, in "Nonlinear Programming" (eds. J. B. Rosen, O. L. Mangasarian and K. Ritter), Academic Press, New York, (1970), 31-66.

[9]

M. J. D. Powell, A hybrid method for nonlinear equations, in "Numerical Methods for Nonlinear Algebraic Equations " (eds. P. Robinowitz), Gordon and Breach Science, London, (1970), 87-144.

[10]

M. J. D. Powell, Convergence properties of a class of minimization algorithms, in "Nonlinear Programming 2" (eds. O. L. Mangasarian, R. R. Meyer and S. M. Robinson), Academic Press, New York, (1974), 1-27.

[11]

M. J. D. Powell and Y. X. Yuan, A trust region algorithm for equality constrained optimization, Math. Program., 49 (1991), 189-211. doi: 10.1007/BF01588787.

[12]

Y. X. Yuan, A review of trust region algorithms for optimization, in "ICIAM99: Proceedings of the Fourth International Congress on Industrial and Applied Mathematics" (eds. J. M. Ball and J. C. R. Hunt), Oxford University Press, Oxford UK, (2000), 271-282.

[13]

H. W. Zhou and Y. X. Yuan, A subspace implementation of quasi-newton trust region methods for unconstrained optimization, Report, ICMSEC, AMSS, Chinese Academy of Science, 2004.

[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), 36-47.

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